Around Caspar Wesse l and the Geometric Representation of

Transcription

Around Caspar Wesse l
and the Geometric Representatio n
of Complex Number s
Proceedings of the Wessel Symposiu m
at The Royal Danish Academy of Sciences and Letter s
Copenhagen, August 1145 199 8
Invited Papers
Edited by Jesper Lützen
Matematisk-fysiske Meddelelser 46 : 2
Det Kongelige Danske Videnskabernes Selska b
The Royal Danish Academy of Sciences and Letters
Commission Agent : C .A. Reitzels Forlag
Copenhagen 2001
Abstract
On March 10 1797 the Norwegian surveyor Caspar Wessel presented an essay On the Analytical Representation of
Direction to The Royal Danish Academy of Sciences and Letters in which he described the geometric representation of complex numbers that has since become standard . The paper was printed in the Academy's Journal two
years later . In order to celebrate the 200th anniversary of this event the Academy arranged a Wessel Symposiu m
on August 11-15 1998 . The contributions to the present volume are based on invited papers presented on that
occasion . Their subjects range over a variety of historical themes related to Wessel and his family, to Wessel's wor k
as a surveyor, to the geometric representation of complex numbers, and to the emergence of hyper-comple x
numbers .
JESPER LÜTZEN
Department of Mathematic s
University of Copenhagen
Universitetsparken 5
DK 2100 Copenhagen Ø
lutzen©math .ku .dk
© Det Kongelige Danske Videnskabernes Selskab 200 1
Printed in Denmark by Special-Trykkeriet Viborg a-s
ISSN 0023-3323 . ISBN 87-7876-236-7
MflVI 46 :2
1
Contents
Preface
3
The Wessel family
The naval hero Peter Tordenskiold & the Wessel famil y
by Hans Christian Bjerg
5
Johan Herman Wesse l
by Bjorn Linnestad
19
Technology Transfer to Denmark
Thomas Bugge's journal of a voyage through Germany, Holland an d
England, 1777
by Kurt Moller Pedersen
29
English instrument makers observed by predatory Dane s
by Dan Ch . Christensen
47
Instrument maker on the run : A case of technology transfe r
by Olov Arvelin
65
Cartography and Astronomy in Denmar k
Wessel as a cartographer
by Leif Kahl Kristensen
81
Heinrich Christian Schumacher - mediator between Denmark an d
Germany ; Centre of Scientific communication in astronom y
by Jurgen Hamel
99
History of Complex Number s
Viète ' s generation of triangle s
by Otto B . Bekken
12 1
Argand and the early work on graphical representation : New source s
and interpretation s
by Gert Schubring
12 5
Inexplicable? The status of complex numbers in Britain, 1750-185 0
by Adrian Rice
14 7
Bellavitis's equipollences calculus and his theory of complex number s
by Paolo Freguglia
181
MfM 46 : 2
2
History of Hypercomplex Number s
Hypercomplex numbers in the work of Caspar Wessel and Herman n
Günther Grassmann : are there any similarities ?
by Karl-Heinz Schlote
20 5
Julius Petersen, Karl Weierstrass, Hermann Amandus Schwarz an d
Richard Dedekind on hypercomplex number s
by Jesper Lützen
22 3
Priority claims and mathematical values : disputes over quaternion s
at the end of the nineteenth centur y
by Tom Archibald
255
Imaginary Elements in Geometry
On the role of imaginary elements in the 19th-century geometr y
by David . E . Rowe
271
MfM 46 :2
3
Prefac e
This book arose as a result of a five day symposium held from the 11th to th e
15th of August 1998 at the Royal Danish Academy of Sciences and Letter s
in order to celebrate the bicentenary of the publication of Caspar Wessel' s
essay Om Directionens analytiske Betegning, et Forsøg anvendt fornemmeli g
til plane og sph(E riske Polygoners Opløsning . In this essay Wessel presented
his famous geometric representation of the complex numbers and made a
generalisation of this analytic formalism to three-dimensional space . Wessel
presented his essay to the Royal Danish Academy on March 10th 1797 an d
it was published two years later in the Academy's journal .
At the symposium two key talks were given about Wessel and complex
numbers : Bodil Branner and Nils Voje Johansen spoke about Caspar Wes-
sel (1745-1818) . Surveyor and Mathematician, and Kirsti Andersen spok e
about Wessel's Work on Complex Numbers and its Place in History . Thes e
talks were based on more extensive papers that have since been publishe d
in volume 46 :1 of these Matematisk-fysiske Meddelelser together with Flemming Damhus's new complete English translation of Wessel's essay1 . Th e
present proceedings contain the majority of the remaining papers given a t
the Wessel Symposium . It can therefore be considered as a supplement to th e
above mentioned book, which in turn embodies the central themes aroun d
which these proceedings turn .
The Wessel Symposium and this book deal with subjects from many differ ent historical fields : literary history, military history, history of technology ,
history of astronomy and geodesy as well as history of mathematics . Such
a variety of subjects may be appropriate in a publication by one of the fe w
remaining academies of both sciences and letters . What binds the different
papers together is their relation to the man Caspar Wessel, to his wor k
as a surveyor, and to the subject of his mathematical essay : complex and
hypercomplex numbers . The style of the papers varies as much as thei r
content . A few papers are conversational in style whereas the majority ar e
more scholarly.
Kaspar Wessel : On the Analytical Representation of Direction . An attempt Applied
Chiefly to Solving Plane and Spherical Polygons . 1797. Translated by Flemming Damhus .
Introductory chapters by Bodil Branner, Nils Voje Johansen, and Kirsti Andersen . Edite d
by Bodil Branner and Jesper Liitzen . Matematisk-fysiske Meddelelser 46 :1 . Det Kongelige
Danske Videnskabernes Selskab, Copenhagen 1999 .
4
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I wish to express my gratitude to the following foundations that supporte d
the Wessel Symposium financially : The Danish Research Councils, Knud
Hojgaards Fond, Norske Selskab and last but not least The Carlsberg Foundation, which has also paid for the publication of these proceedings . I als o
wish to thank Pia Grüner and Flemming Lundgreen-Nielsen at the Roya l
Academy for their help with the project and the authors who produced goo d
scholarly works and conscientiously refereed each other's papers . Finally I
am obliged to Jan Caesar at the Department of Mathematics, Copenhage n
University, for having produced a nice and uniform LATEX layout of all th e
contributions and to Knud Sorensen, Aarhus University, for his careful correction of the language .
Jesper
Lützen
MfM 46 :2
5
The Naval Her o
Peter Tordenskiol d
the Wessel Family
Hans Christian Bjerg *
1 The Wessel Family
The Wessel Family has contributed in several ways to the historical develop ment of the Double-monarchy of Denmark-Norway which ended in the yea r
1814 when the two kingdoms separated . This volume is primarily dedicate d
to an example in the area of science, and it also contains a contributio n
about Johann Herman Wessel, one of our famous poets . And in the Danis h
and Norwegian military history you will also find the name Wessel . One
of our most famous naval heroes, Peter Tordenskiold, was christened Pete r
Wessel and was a true member of the mentioned family . This paper wil l
deal with this man and his merits in the beginning of the 18th Century .
Everyone in Denmark and Norway knows the name Tordenskiold . Several
anecdotes are told about him, but his fame is probably mainly due to th e
fact that during this century his portrait has been reproduced on the mos t
sold matchbox in Denmark .
But first a little about the Wessel Family . In the middle of the 18th Centur y
the family was of the opinion that they had immigrated from the Nether lands, and indeed there is a Dutch locality called Wessel . But it is a fac t
that in the 16th Century one finds the family name Wessel in a couple of th e
* Chief Archivist, The Military Archives, The Danish National Archives, Rigsdagsgården, DK-1218 Copenhagen K, Denmark .
6
H. C. Bjerg
MfM 46 : 2
Hanseatic cities in the Northern part of Germany, where the trade agent s
had good connections with Norway, especially with Bergen, where Germa n
merchants occupied a part of the city . It is also possible that the Wesse l
family could have immigrated this way . We find for the first time a Wesse l
from the Netherlands in Bergen in Norwegian files from 1593 .
In any case, around 1620 a man with the name Jan Wessel lived in Bergen ,
and the grandfather of the naval hero Peter Tordenskiold named Henri k
Jansson Wessel is traced in the archives from the middle of the 17th Centur y
as a merchant and citizen of Bergen . Henrik Wessel's oldest son, Jan Wessel ,
was born in 1646 and went as a grown-up to Trondheim, a town in Norwa y
north of Bergen . He married a very young girl, born 1656, Maren Scholler ,
who was out of one of the rich families in Trondheim . As a 16-year-old gir l
she already gave birth to her first child . Within the next 26 year she becam e
a mother of 18 children
12 sons and 6 daughters . An anecdote tells that
at an advanged age Jan Wessel was on a trip sailing with a merchant shi p
to Spain . The ship was boarded by French privateers . Investigating the
passengers they were told that Jan Wessel was a father of 18 children . The
privateers were so impressed with a man of such a "capacité" that they left
the ship without doing any harm .
Jan Wessel owned some buildings in Trondheim and some ships and wa s
as a whole a respectable citizen . He became what we can call an alderma n
and a member of the Council of the City in 1693 . His Bible is still kept i n
Norway and on the front page of the book we can follow the growth of th e
Wessel family. Every new child was carefully listed .
Son number 10 was Jan Wessel named after his father . He went to sea and
became a chief pilot in Norway . This man became the grandfather of Caspa r
and Johann Herman Wessel . Peter Tordenskiold was in this way granduncl e
to Caspar Wessel .
Of the 6 daughters 2 died very young, 3 married merchants in Trondheim ,
and one married a vicar . Of the 12 sons 4 were educated as clergymen, 1
died young, and the rest went to sea . Of these, 2 died in the Dutch navy,
1 became captain in the Russian Navy and 2 became viceadmirals in the
Royal Danish Navy, "but only one of these was Tordenskiold" - as an ol d
song goes .
Child number 14 was Peter Jansen Wessel . He became a naval officer jus t
when the Great Northern War 1709-1720 broke out . In this war Sweden's
MfM 46 : 2
Peter Tordenskiold
Christoffer W.
Jan W .
1684-172 9
chief pilo t
1674-1754
vicar in BjørnØr,
Norway
Andreas Johan Christopher
Bredal W . W . Tordenskiol d
1714-98
vicar in
Akerø
1727-9 3
officer ,
superintendent ,
enobled 1761
Peter
Joha n
Schnitler T . Christian T .
1767-1827
1778-182 8
captain second lieutnant ,
last member of
the noble family
Karen W .
Ole W .
1685-cä .1756
1687-1748
Knud Brun
vicar in Vestby
alderman in Trondheim,
founder of the family
Wessel-Brow n
Jonas W.
1707-85
rural clean,
vicar in
7
Peter W .
Casper v . W.
ennobled
1693-176 8
Tordenskiold naval officer ,
1690-1720
enobled 172 0
viceadmiral
Joachim W .
1739-9 1
officer
Vestby
Johan
Herman W .
1742-1785
poet
Ole
Caspar W .
Christopher W .
1745-181 8
surveyor
1744-179 4
deputy in "norske
mathematician
generalitets- og
ko romis s ari at skollegi um"
Figure 1 : The genealogical tree of the Wessel family
neighbours tried to reduce the Swedish dominance in the Baltic area . Th e
Danish king was primarily interested in getting the former Danish area s
in the western part of Sweden back to Denmark . Because of his merit s
Peter Wessel was ennobled in 1716 with the name TORDENSKIOLD . It means
thunder-shield or freely translated Thunderbolt . On that occasion he got a
coat of arms which illustrates his profession and background . He was a
thunder for the enemy and a shield for his king and country as an ol d
interpretation of the name goes .
Caspar Wessel (1693-1768) was Peter's younger brother . He was promote d
to viceadmiral, which allowed him to marry a rich widow and get rid of hi s
great debt . However, he was not as strong a character as Peter and wa s
promoted as a result of his older brother .
I have mentioned a couple of names from the Wessel family and maybe yo u
have observed that the family re-use their Christian names . This tradition
S
H.C. Bjerg
MfM
46 : 2
Figure 2 : Tordenskiold's Coat of Arms . 1 .) Shows the symbol of lightning an d
thunder on a blue background referring to the name Tordenskiold . 2 .) Shows a
white eagle on a red background referring to Tordenskiold's capture of the Swedis h
warship VITA ORN in 1715 . 3 .) Shows two cannons and three cannon balls on a
red background symbolizing the three gunshots which were the special mark fo r
Danish warships in former time . 4 .) The Norwegian Lion Statant in gold on a re d
background refers to the Danish-Norwegian nationality of Tordenskiold . It is the
Norwegian coat of arms in the colours used in the Danish coat of arms .
NIfM 46 :2
Peter Tordenskiold
9
has continued up to these days .
2 Peter Wesse l
We have a lot of anecdotes and stories about Peter Wessel as a child . He wa s
apparently a wild boy according to the first biography about the Danish Norwegian naval hero published 20 years after his death . By nature he wa s
an adventurer and he stayed often by the harbour and on board his father' s
ships . We know that as a 14-year-old boy he arrived in Copenhagen . Indee d
at that time he visited one of his father's friends in the Danish capital ,
Dr . Peder Jespersen, who was then a spiritual adviser to the Danish Kin g
Frederik IV, a man of importance . Peter Wessel stayed in the house of Dr .
Jespersen, who wrote to the parents in Trondheim that their son was now i n
Copenhagen . According to information in the archives Peter ran away fro m
home without giving any messages to his parents . We do not know how h e
made his way to distant Copenhagen . The Danish king made a voyage t o
Norway in 1704 and passed Trondheim on that occasion . We suppose that
Peter Wessel joined the king's retinue one way or another and followed th e
company back to Denmark .
Peter Wessel' s passion was the sea . Through Dr . Jespersen Peter Wessel had
had the opportunity to meet the King, and in 1706 he applied to the Kin g
to become a midshipman in the Royal Navy . The Royal Naval Academy had
been established in 1701 in Copenhagen as one of the first of that kind i n
Europe . However, there was no place vacant at that time and Peter Wesse l
was urged to get practical experience as a sailor . He therefore signed article s
on board a ship going to Africa and the West Indies as a Slave transport .
He was on board this ship until 1708 . In 1709 he became a midshipman ,
but at that time he was on his way with a new ship to East India so he di d
not get the message about his new position until he returned to Bergen i n
1710 .
The war with Sweden broke out in 1709, and the need for naval officers wa s
very acute . Peter Wessel therefore got only a very brief and quick training a t
the Naval Academy . In spite of that he was already an experienced navigator ,
and in 1711 he was promoted to lieutenant and worked as the second i n
command on one of the small frigates . He then was detailed for duty in
10
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MfM 46 : 2
the coastal navy in Norway. In 1712, when he was 21 years old he became
captain of a small frigate called LØWENDAHLS GALEY of 20 guns .
At that time his superiors began to observe Peter Wessel's special talent fo r
naval warfare and tactics . To understand this fact it is necessary to explai n
a little about the naval tactics practised in the very beginning of the 18t h
Century.
The armoured square-rigged ship became common from the beginning o f
the 17th Century as the normal platform in naval warfare . The technology
of this type of ship indicated that the guns were placed along the sides of th e
ships and that the stern and the bow were practically without any guns .
The warships of that time fought in the so-called line ahead in order t o
maximise the shooting angles of the guns and to minimise the dead angle s
at the stem and at the bow . At the end of the 17th Century there wer e
developed two different tactical ways to use the warship . The Mêlée tactic s
and the formal tactics . In the beginning of the 18th Century the forma l
tactics were accepted by all great naval powers as the correct way to make
warfare at sea . The drawback of the formal tactics was that the battles a t
sea stagnated . The formations of the ships became too defensive and it wa s
difficult to destroy the enemy. Yet, in all the naval academies at that tim e
formal tactics were taught as the only one and it was the only one whic h
was allowed according to the fighting instructions and the rules .
As I mentioned, Peter Wessel's stay at the naval academy was too shor t
to indoctrinate him with the lessons and rules of the time . He saw tha t
the best way to attack the ships of the enemy was to attack them in thei r
dead angles, which was not the normal way to do it . His tactics were very
surprising for the opponent, and as a consequence were very successful . I n
short, Peter Wessel followed his own tactical instinct instead of rules give n
at the academy.
In 1712 he was called on duty in the Main Fleet of the Danish Navy which
operated in the Baltic Sea against the Swedish Navy. His efforts were remarkable and he became famous for his risky but effective way to do scou t
raids for the Admiral in order to gather intelligence about the enemy . As
an award he was promoted to lieutenant commander jumping over 51 othe r
lieutenants placed before him on the Officers' Roll . In 1713 he worked i n
the Norwegian Sea and was hunting Swedish merchant ships and privateers .
His results were remarkable .
MfM 46 :2
Peter Tordenskiold
11
Figure 4 : A portrait of Peter Tordenskiold as a Vice-Admiral, painted in the yea r
171 9
In July 1714 Peter Wessel observed a ship under British colours . He himsel f
sailed under Dutch colours! The other ship struck the colours and flew th e
Swedish flag . Peter Wessel then showed the Danish colours . The other frigat e
was bigger than Wessel's, and according to the Danish fighting instruction
12
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MfM 46 : 2
it was not allowed to fight with ships superior to one's own . Nevertheless ,
Peter Wessel continued the fight . During the fight the other ship was heavil y
damaged, and Peter Wessel decided to fly the white flag, the flag of truce ,
communicating that he wanted to talk with the captain on board the other
ship . Wessel sent out a small vessel with a negotiator, who brought greeting s
from Wessel . The negotiator told the enemy captain on behalf of Wessel tha t
the Danish ship had only very few cannonballs left, and therefore he woul d
ask the enemy captain if he could lend him cannonballs so that the fight
could go on . To this surprising question the enemy captain answered that h e
only had what was necessary for himself, but that he would like to present a
toast to the Danish captain . After Wessel's negotiator had returned to th e
Danish ship the two enemies sailed very close to each other, toasted over th e
bulwarks of the ships and asked each other to give friends in Copenhage n
and Gothenburg respectively their regards . Through this manoeuvre Wesse l
succeeded in gathering information about the enemy. The Swedish ship wa s
bought in England and was on its way to Gothenburg under the comman d
of a British captain .
When he returned to his base in Copenhagen, Peter Wessel wrote a complet e
report about the fight and told about the message to the enemy captain an d
the toast afterwards . Some of the members of the Danish board of Admiralt y
became very angry reading this report . From their point of view Peter Wesse l
had uncovered his own weakness in front of the enemy, a weakness which th e
enemy captain could have used to take Peter Wessel's ship . The young naval
officer had offended against the Naval Articles and he was court-martialled .
The point here is that we are not dealing with an anecdote, but with facts .
All the papers in this case can be found in the archives . In the files we
can follow the court-martial . Wessel explained that the enemy ship was s o
damaged that there was no risk for him to go closer to the other ship . On th e
other hand the ship was too superior to board . He had sent the negotiato r
over in order to get information about the ship because it was not know n
in the theatre before . Peter Wessel was cleared off. Just after the acquitta l
he went to see the king, who promoted him to Commander in the Navy. H e
was 23 years old .
It is not surprising that he was already envied at that time . In a letter he
wrote that he was not looking for a higher charge but for more responsibility .
It was the challenge and its realization which fascinated Peter Wessel .
1715 became a crucial year for the young naval officer . He served in the
MfM 46 :2
Peter Tordenskiold
13
main fleet in the Baltic Sea as a scout for the Admiral and a cruiser captain .
Based on intelligence about the Swedes fleet provided by Peter Wessel it wa s
decided to let the Danish fleet attack in April . The result was the Battle o f
Fernern, where the Swedish suffered a defeat . With his small frigate Pete r
Wessel was very active during the battle . He and his crew boarded a heav y
and new Swedish frigate called VITA ORN, The white Eagle, and prevente d
the Swedish captain from destroying his own ship . Afterwards Peter Wessel
observed that a group of enemy ships, which at first had escaped fro m
the battle, had been grounded by its own crews . One of the ships was the
Swedish flagship with the enemy admiral on board . Wessel sailed straight
to the flagship, took the admiral prisoner and ordered him to stop the self destruction of his ships .
As a reward for an excellent effort Peter Wessel got the command of th e
conquered Swedish frigate, which was incorporated in the Danish-Norwegia n
Navy under the name HVIDE ORN . In October of the same year he opene d
battle with a ship-of-the-line and a heavy frigate, which escorted a convoy i n
the Baltic Sea . In spite of the Swedish superiority Peter Wessel succeeded ,
thanks to his excellent tactical talents, in damaging the two ships seriousl y
and he took a great part of the convoy as prizes .
After this event reports were sent from the fleet to the Admiralty in whic h
Peter Wessel was praised for his tactical skill . In February the following year
1716 came his ennoblement .
3
Tordenskiold
When you study the history of Tordenskiold you will discover that the realit y
is often more fantastic and incredible than the myths and the anecdotes .
One evening when Tordenskiold was dining at court, the king expressed th e
wish to know the opinion on the on-going war among the former Danish
subjects in Sweden . After the royal table Tordenskiold went out to Holmen ,
the naval base, appointed a crew to a gunboat, and sailed to the Swedis h
coast during the night . Very early in the morning he reached the enem y
coast . The two nearest buildings turned out to belong to a vicar and th e
local coast Watch officer . Tordenskiold woke up the vicar and forced hi m
to follow him back to the boat . A servant, a young boy, woke up and wa s
forced to go to the boat too . The officer suffered the same fate as the vicar .
14
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MfM 46 : 2
All three then sailed during the sunrise to Copenhagen, and were presente d
to the king during his breakfast . The king was very amused with Tordenskiold's crazy idea . He talked with the Swedish prisoners and gave them gifts .
Afterwards Tordenskiold brought them to the Naval Base where the vica r
and the officer were cross-examined . The vicar was an elderly man and could
remember when the western coast of Sweden belonged to the Danish king .
He did not mind telling all that he knew about the Swedish king, the arm y
and the contemporary political conditions in Sweden . All the questions an d
answers were noted in a file . The officer refused to tell about the condition s
in Sweden . The following night the three Swedes were brought back to thei r
home by Tordenskiold's crew .
During the following days the Swedish authorities interrogated the vicar an d
the officer about their involuntary trip to Copenhagen and about the publi c
conditions in the Danish capital . The vicar and the officer both told tha t
the Danish Naval authorities had tried to interrogate them about Swede n
but that they had told nothing . Actually the vicar said that his lips ha d
been sealed during the interrogation .
This story is not a myth either . In the Swedish National Archives they keep
the files about the Swedish inquiries, and in the Danish National Archive s
we have the protocols belonging to these interrogations in Copenhagen .
On a later occasion Tordenskiold amused the king in a similar way . Again
he went to the Swedish coast . In the nearest house a wedding was bein g
celebrated and Tordenskiold and his crew so to speak kidnapped the newl y
married couple and took them to Copenhagen and presented them to th e
king, who gave them gifts . They were also brought back to their home . Th e
files do not tell us whether or not the wedding continued after the return o f
the kidnapped bride and groom .
Tordenskiold repaid the coat of arms in June 1716 . A part of the master pla n
of the Swedish King Charles XII was to take Norway from the Danish king .
In the middle of 1716 he began a campaign against Norway . To support his
advancing army Charles XII ordered a big transport fleet with supplies along
the Swedish western coast to meet the army near the border to Norway . Th e
success of the campaign depended on this fleet .
At the end of May 1716 Tordenskiold investigated the Swedish coast sout h
of Gothenburg when he got intelligence about the mentioned transport fleet .
He followed the fleet and found out that it had anchored in a small and very
MfM
46:2
Peter Tordenskiold
15
narrow creek called Dynekilen . The entrance was only 100 metres wide . The
Swedish officers were so sure that this place was safe that in the evenin g
they went for a party on shore .
Tordenskiold had no orders to attack, and the correct behaviour woul d
have been to get such orders from his superior . But he evaluated that th e
surprising moment would disappear if he waited for orders . The wind was
favourable and he went into the creek and took a part of the enemy transpor t
ships as prizes and destroyed the rest . This operation had a major influenc e
on Charles XII's war campaign against Norway . The Swedish Army go t
no supplies and the army was forced to withdraw to Sweden . Dynekilen i s
considered the greatest and most important of Tordenskiold's merits .
During the year 1717 Tordenskiold again operated in the Baltic Sea but wa s
not very successful . He also attacked the fortresses of Gothenburg withou t
luck . The following year he commanded a scout squadron based on Christiansø with the task of observing the movements of the Swedish Fleet base d
in Karlskrona . In the last months of 1718 he was called to the Norwegian
theatre of War again because of a message about a new campaign agains t
Norway . In December he heard as one of the first commanders about th e
death of Charles
XII at Frederiksstad . One of Tordenskiold's advantage s
was his talent for a permanent gathering of intelligence . He was very much
aware of the importance of the death of the Swedish king for the whole war .
He therefore immediately sailed on a fast-going vessel to the king in Copenhagen . Tordenskiold was the first to bring the message to the king, and a s
a reward he was promoted to rear-admiral on the spur of the moment .
In July 1719 Tordenskiold began to lay siege to the important fortress Karlsten at Marstrand on the Swedish west coast . After a month the fortres s
was captured without any fighting and due to a trick - very typical o f
Tordenskiold . An anecdote tells us that Tordenskiold in the disguise of a
fisherman went into the enemy city, sold fish and interrogated the citizen s
about the troops in the fortress . The troops were German mercenaries .
In Denmark we have a special phrase : "Tordenskiolds soldater" (The soldier s
of Tordenskiold), meaning that it is the same people you se all over : in th e
media and in the boards of societies and companies etc . This phrase comes
from Tordenskiold's siege of Karlsten-Marstrand . It tells that Tordenskiol d
invited the commander of the fortress to inspect his troops which were line d
up in the city streets below the fortress . The commander went through al l
the streets in the town and everywhere he saw troops lined up . He realised
16
H. C. Bjerg
MfM 46 :2 .
that he did not have a chance to resist Tordenskiold and so he decided t o
surrender under the condition that all of his troops were allowed to leav e
the fortress unharmed . In reality Tordenskiold's limited troops after the
inspection by the Swedish commander ran around the corner and were line d
up in another street where the commander then inspected the same troop s
for the second or the third time .
We know in fact that the commander was not really tricked by Tordenskiol d
on that occasion but was afraid of the reinforcements which he knew wer e
coming up from Denmark .
After the capture Tordenskiold was promoted to viceadmiral and member
of the Admiralty. As a supplement he received a portrait of the king frame d
in brilliants . This portrait was only given four times in the reign of Frederi k
IV .
When the peace came in 1720 Tordenskiold was restless . He was a typica l
man of war, so he planned a journey to Germany, France, England, an d
maybe also to Russia to look for new challenges and offer his naval expertise . In Hanover in Germany Tordenskiold was involved in a quarrel with a
Swedish-Lithuanian colonel . It ended in a duel in which Tordenskiold wa s
killed on the 12th of November 1720 . There have been a lot of discussions
about how the duel was settled but we shall not go into the details here .
4 Remembering Tordenskiold
The body of Tordenskiold was brought home, but there were difficulties
with the funeral because according to Danish law duels were forbidden, an d
victims of duels were therefore not allowed to be buried in consecrated earth .
In spite of his great and many merits for the country many important and
envious persons blocked a funeral ceremony of the great naval hero . He wa s
placed in a coffin in the cellar of Holmen's Church in Copenhagen where h e
rested until 1819 when the Danish king Frederik VI erected a sarcophagu s
for his body .
In 1864 a statue of Tordenskiold by the sculptor H .W .Bissen was erected
in front of the Cathedral in Trondheim in Norway. Later, in 1879, a similar
statue was erected near Holmen's Church .
Already during his lifetime Tordenskiold was the subject of several publications . Since his death nearly 1,000 books and articles have been published
MfM 46 :2
Peter Tordenskiold
17
about the naval hero . And the bibliography still gets new titles about Tordenskiold . A book with all his letters - around 1,200 written in th e
service has been published .
The administration of his estate has its own history, because it became
the longest and the biggest in Danish history . It took 24 years to carry it
out . The State owed Tordenskiold a lot of money for the many prizes h e
took during the war, but he didn't leave any heirs, so the State was no t
interested in giving the many brothers and sisters the money . Therefore th e
state authorities tried to avoid its obligations . As for the family, they ha d
a lot of quarrels over the money and other values left by Tordenskiold .
On Tordenskiold's sarcophagus in Holmen's Church is engraved the name s
of his victories : "Dynekilen - Marstrand - Elfsborg" together with the fol lowing inscription :
HIS KING MENTIONED HIM WITH HONOUR HIS ENEMIES WITH FEA R
THE ANNALS OF DENMARK KEEP HIS MERIT S
IN REMEMBRANC E
FREDERICH VI ERECTED THIS SARCOPHAGUS TO HIM .
5 The Chronology of Peter Tordenskiol d
1690, 20 October, Born in Trondheim, Christened as Peter Janse Wesse l
1704-06 In Copenhage n
1706, 28 March, First application to the King in order to become a midshipman
1706-08 On board a slave ship bound for Africa and the West Indie s
1708-10 On board a merchant ship bound for Tranquebar, Indi a
1709, 11 January, Received into the Naval Academ y
1709-20 The Great Northern War
1711, 17 July, Sub-lieutenan t
1712, 14 October, Lieutenant-Commande r
1714, 28 December, Commande r
1716, 24 February, Ennobled as Tordenskiold
MfM 46 : 2
H. C. Bjerg
18
1716, 8 July, The Destruction of the Swedish Fleet in
Dynekilen
1716, 18 July, Captain in the Danish Royal Navy
1718, 30 December, Rear-Admira l
1719, 25 July, The Capture of
Marstrand Fortres s
1719, 17 August, Vice-Admiral and member of the Admiralt y
1720, 7 September, Leaving Denmark for a trip to Germany, France an d
Englan d
1720, 12 November, Death of Tordenskiold in a duel in German y
1819, 17 July, Unveiling of the sarcophagus in Holmen's Church
1879 Unveiling of the Statue of Tordenskiold at Holmen's Church
MfM 46 :2
19
Johan Herman Wesse l
Bjorn Linnestad *
I would like to start with the back scene, so to speak, for the performance
of the Wessel brothers in Copenhagen .
The four hundred years' history of the Danish-Norwegian twin monarch y
has been, and still is, a mental complex to many Norwegians . The unite d
kingdoms were the result, not of conquest and war, but of legal heritage . I
think this background is important for the understanding of the Norwegians '
position in eighteenth-century Denmark .
It was not a disadvantage at that time to be Norwegian born . The Jutlanders
and the Norwegians were both to some degree alien to the Copenhagener s
with their different language and manners . However, there was a difference
in how the two provinces were represented in the capital : Jutland broadly
and unselected, the whole scale of persons from the manual worker an d
the maid to the tops of society. Norway on the other hand was selectivel y
represented, mainly by two groups, the university men and the guardsmen
- both pick of the crop, either intellectually or physically .
It has also been claimed that the Norwegian in general had a wider horizon ,
geographically, than the Dane . Denmark's income was based on inland farming, while Norway for a large part made a living of trade and shipping . The
Norwegian, if not the average, knew London and had a cousin in Bordeaux .
The Jutlander represented to the Copenhageners simplicity, the Norwegia n
complexity. However, as Johan Herman Wessel so tolerantly put it : "we are
all Jutlanders to our Lord . "
At the time of the Wessel brothers, Copenhagen had 50,000 inhabitants, and
the university men were clearly visible in the town . Around 1770 the literarily interested Danes made their circle around the poet Johannes Ewald ,
* Medisinsk Afdeling, Sykehuset Østfold Askim, N-1800 Askim, Norway
B. Linnestad
20
MfM 46 : 2
and the Norwegians gathered around Johan Herman Wessel . The former
group cultivated pre-romanticism, the latter the rational tradition from th e
Greeks and the Romans . Both circles willingly leaked their wit and irony
to the outer public, and the group around Wessel had not the minor admiration . The literarily conscious Copenhageners used Norwegian words an d
expressions to signal that he was up to date .
The Norwegian literary circle became formally Norske Selskab
the Norin 1772, and through the following years it was constantly
wegian Society
watched over by the authorities . This group of young idealists could represent the seed of Norwegian independence . The members of the Society ,
however, were in general loyal to the King, but they worked for more extensive home rule and a Norwegian university .
The Norwegians were, like today, very patriotic . They sang "For Norge ,
kjempers fØdeland" - "To Norway, birthplace of giants" - written b y
Johan Nordahl Brun, a later bishop of Bergen . Johan Vibe, Claus Fasting ,
Niels Bredal were the other well known names . Whether they were student s
of law, of theology or something else, they were all writers, poets and quit e
a few also playwrights . In sum they offered a substantial impact on Danis h
society.
From the beginning of the 1760's the four Wessel brothers from Vestby
in Norway were gathered in Copenhagen, and inside the circle that late r
became Norske Selskab . Jonas, the eldest, had low ambitions :
Ett jeg vet for viss t
jeg vil blive copiist
In my translation :
All I want, in this I'm star k
I shall be a common clerk
Those two lines are, to my knowledge, his complete works - and he succeeded, he became a clerk in the Ministry of Finance . The three younge r
brothers, Johan Herman, Ole Christopher and Caspar all did excellently a t
the University. Strangely enough, Johan Herman broke off after two lowe r
grade exams . His knowledge of Hebrew had not been convincing, but apar t
from that he got top marks in physics, arithmetic, geography, metaphysics ,
history, logic and Latin . He was obviously broadly gifted .
Why Johan Herman stopped in his academic career has never been satisfactorily explained . It could be due to a shift in mood and a longing for
MfM 46 :2
Johan Herman Wessel
21
isolation . His life during the following ten years is very obscure . The only
thing known to posterity is that he taught himself the living languages ,
English and French, later in life also Italian and Spanish .
Ole Christopher became an outstanding capacity within the law . Caspar
also became a lawyer, but first and foremost an excellent topographer an d
mathematician . Johan Herman Wessel, however, is my main topic .
There are some mysterious or at least strange features in this man's lif e
and in our apprehension of him . I am going to focus on three questions an d
thereby I hope to throw some light on the fascinating Johan Herman .
1)
Why is so little known about him?
2)
Why has he, as a writer, survived his contemporaries ?
3)
Why did this brilliant poet produce so little ?
It is a striking fact that very little is known about Johan Herman Wessel i n
the biographical sense . He himself wrote nothing in that respect apart fro m
a few ironical lines . Here I have to put in an apology : We have, if any fe w
translations of Wessel's work . His expression is cunning with many shades ,
and translation is difficult . If I were to try to put him into English words, I
would certainly not convince you of his linguistic dexterity . I have therefore
chosen to quote him in his own language and I ask English-speakin g
readers to excuse this .
Han syntes født til bagatelle r
og noget stort han ble ei heller
He thought himself born to trifle away his life - and subjectively so he did .
Why then did his contemporaries not write about him? After all he wa s
surrounded by academics with pens . And he was the most prominent o f
his circle, the "Blossom and Crown" in the Danish poet Oehlenschläger' s
words . This reserve could be due to Wessel's overt alcoholism and represent a
considerate discretion . In addition, with the 1770's came a shift in attitude s
in the Danish-Norwegian society . The previous decades had been libera l
in Copenhagen as in the rest of Western Europe : Rousseau's philosophy
and writings, Henry Fielding's novels, and John Cleland's `"Fanny Hill" .
Copenhagen had a lag, but in 1746 the puritan king Christian VI die d
and was succeeded by his son Frederik V . The grave and stiff life at cour t
changed into orgies, and under the next king, the mentally insane Christia n
VII, it grew worse still . Dr . Struensee became in reality head of State . He
B . Linnestad
22
MfM 46 : 2
liberalized society, but also made the Queen pregnant . It had to end in a
tragedy and Struensee was executed in 1772 . It took 200 years before the
state of Norway the next time appointed a medical doctor prime minister .
The reaction to the dramatic events came, not only politically but als o
morally. To enjoy Wessel grew, at least officially, out of fashion .
Another fact difficult to explain is Wessel's popularity through more tha n
two centuries . He presented no deep thoughts, he shunned topics more or les s
compulsory to poets : the praise of God, the praise of Nature, the praise o f
Women and Love . Unlike his fellow poets he avoided patriotism and heroism .
Why then is he quoted more often than any other poet of his time? I thin k
the answer simply is his mastery of sentences . He often used the alexandrin e
verse with the same number of syllables in each line and a rhyme at the end .
In "Kierlighed uden Stomper" - "Love without Stockings"
it sounds like
the title of an erotic movie today - we are told that Johan the tailor ha s
been away to mend the trousers of a major, but one can suspect additiona l
activities . On his return he addresses his girlfriend Grethe with the smooth ,
elegant and flattering alexandrine lines :
Hvor ser jeg glad ige n
de mange yndighede r
som den må være blin d
der kan ej se hos Ede r
In my translation :
Now glad again I see
the many charms of you .
The blinded only doubt s
that this is fully true
As I said, Wessel avoided the traditional topics of a poet, God, Natur e
etcetera, with very rare exceptions . Wessel's God is a tolerant power :
Det sanne vel og ve er frukt av dyd og last
og denne sannhet står til Himlens ære fas t
True happiness is based on a combination of virtue and vice and Wessel' s
God has accepted this .
Born in a rural community in Norway, Wessel became ultra-urban . He rarely
and unwillingly left the streets of Copenhagen . Once in spring, however ,
his friends had persuaded him outside the town walls to make homage t o
flourishing nature . The result was two of Wessel's most wonderful lines :
MfM 46 :2
Johan Herman Wessel
23
Og så, den grønne mark, den var
ja, Herregud
man ved hvordan en mar k
når den er grø n
ser u d
In my translation :
Look
The meadows, the fields .
Oh, good old Lord .
The nature yield s
what the eyes affor d
Even if nature was delightful, it deserved no more than a couple of line s
from him .
So to the field of love and eroticism . He avoided those themes, the historian s
of literature tell us . Israel Levin, the great connoisseur of Wessel, said tha t
in all his writing we do not find a vulgar word, not an obscene thought, hi s
Muse was entirely pure . Levin worked and wrote in the Victorian era, an d
I think he has voluntarily overlooked a little poem : Let others enjoy gold ,
Wessel himself found Heaven in the crotch of his girl .
La andre tenke, sige
Gull giør oss lykkelig e
jeg fant mitt Himmerig e
min pig e
i ditt skjød
It was to me unthinkable that a young man surrounded by other young me n
should not cultivate at least the conception of eroticism . So I started my own
hunt, simply to try to rehabilitate Wessel as a normal, healthy man . In th e
archives of the University Library of Oslo I found handwritten notes afte r
Gregers Lundh, the first professor of economic history in Norway . He wrot e
a never published "Contribution to a Collection of J . H . Wessel's Obscen e
Poems ." I shall not give you the most vigorous verses, only a few lines tha t
describe the little `"thing" a girl possesses :
En pige er en ting
som nok en ting beskytte r
og uten denne tin g
hun mannfolk ikke nyter
B . Linnestad
24
MfM 46 : 2
og vil man i den ting
en annen ting innføre ,
fremkommer der en ting
som nye ting kan gjør e
It is an old speculation why this brilliant and gifted poet did not produc e
more than he did . His complete works constitute little more than one volume . Wessel described himself as lazy :
Han åt og drakk
var sjelden glad
sine støvlehele gikk han skjeve
Han ingen ting bestille gad
tillist han gad ei heller leve
At last he did not even bother to live .
He also compared himself to his diligent brother Caspar :
Han tegner landkort og leser love n
er lig'så flittig som jeg er doven
For several reasons I do not believe in the "laziness theory" . There must
have been another explanation for his scarce production .
When Johan Herman Wessel died 43 years of age, he left an impressive debt ,
a widow and a son he doubted was his own . Then his friends publishe d
his complete works to support the widow with the income . In the preface
Christen Pram regrets that his gifted friend did not write more than h e
did . Neither he nor the other contemporaries can explain this . Pram think s
Wessel was hampered by ill health, and tells that the poet was bedridden fo r
more than a year due to a painful fistula in the jaw . This fistula must have
been caused by a tooth infection . Wessel suffered from toothache which h e
described as the Devil's stand-in on earth :
Du er på Jorden vise-Fande n
og halve værre enn den anne n
However, once a fistula opens, the pain goes . Christian Pram's explanatio n
was wrong . What else makes a 34-year-old man go to bed and stay there
for a year? It could be a palsy, an orthopaedic problem or heart or lun g
disease . However, all these conditions give symptoms easy to observ e
and no such observations were made . What to look for then? My diagnosis
MfM 46 :2
Johan Herman Wessel
25
is a grave and deep depression, most likely the genuine melancholy seen i n
persons suffering from manic-depressive disease . Do we have other facts t o
support this? Yes, certainly . Shortly before his long stay in bed Wessel ha d
written his only two serious poems "Ode til Søvnen"
"Ode to sleep" an d
"Nøisomhet" - "Contentment" . In many verses he overtly describes hi s
way into darkness :
Som jeg den halve klode var
innsvøpt i mørkhets tause tåge
og på de lukte øyenlåge din grumme broders bilde bar
On the back of his eyelids he saw the picture of death the grim brothe r
of sleep . In another verse he indirectly tells us that he has experienced th e
same before . He knows that light will come back :
Men ilden ulmer, livet gror
Du snart igjen skal krefter vinn e
Min ånd du snart igjen skal finn e
en morgen skjønn, din Skaper stor
I shall not tire you with all the pieces of the puzzle that in my mind document his deep depression . But what about the maniac phases if he wer e
manic-depressive? His "ups" must most likely have been less pronounce d
than his "downs" and therefore less obvious . I find it however borderin g
on the pathological to write a masterpiece like "Love without stockings" i n
six weeks . Maniac phases can also manifest themselves as quarrels difficul t
to understand . Now we have to remember that Johan Herman was a shy
person, very polite and very considerate in all his being . A couple of year s
after "Stockings", perhaps he was still "upgoing", Wessel published "Herremanden" : a squire dies and ends up in Hell because he has exploited his
peasants :
En herremann sov engang he n
og så skal alle herremenn
hvor gjerne de enn leve ville
Og det er ill e
å dø når man ennu ei vill e
In hell the squire is very astonished to meet his personal servant Jochum . It
turns out that Jochum had made the squire's wife pregnant and therefore
ended up underground .
B. Linnestad
26
MfM 46 : 2
In Copenhagen the rumours started : the poem was to be read as an obituary
of the late general Nergaard . The general had a daughter and two sons, on e
if the poem really had a concret e
of whom was accused of adultery
address . After some hesitation the son attacked Wessel crossly in the paper ,
and the provoked poet answered with a proposed epigram for the squire' s
tombstone :
Her hviler Krigsraad Nergaard
deri store, store, store
Kun Himlen vet hva godt han
gjorde, gjorde, gjord e
At tale om hans kjære fru e
så var hun vennlig som en due
At tale om hans tvende sønne r
det ei urnaken lønner
Men når jeg tenker på hans datte r
kan jeg ei bare meg for latte r
Here rests general Nergaard . If he ever did a good thing, only Heaven knows .
Not only Nergaard but also his wife, sons and daughter are exposed t o
disgrace . The mentally sound Wessel would never do such a thing .
On another occasion, we are now in the 1780's, Wessel also acted very
inconsiderately and very unlike himself. At that time he rented a smal l
flat in Fiolstræde where he lived with his wife and son . The landlady ha d
the habit of steadily creating rows with everybody around her Wessel
included . Then the landlady died, and a collective sigh of relief must hav e
been heard . Wessel owed her money, but he was broke . The family of the
deceased then suggested that instead of payment he should write an epigra m
for her grave . From a famous poet that gave prestige woth more than money .
Wessel had no money and therefore no choice and he presented the lines :
Hun er død, hun er begravet
Jeg har fred, og nabolavet
In my translation :
Now she rests in the coffin's woo d
I'm at peace like the neighbourhood
Johan Herman Wessel
MfM 46 :2
27
It certainly was tactless, no matter who the lady was .
The Johan Herman known by his friends had a mild, intelligent irony with out any aggression . Once a pompous officer said to Wessel : "If I had a stupi d
son, he should become a poet ." The answer came immediately : "Your father
obviously had another opinion . "
One could suspect that Wessel's premature death was linked to his psychiatric condition, but that was not the case . During his last months he was
in a good mood, and it was an accidental infection that killed him . Nerve
fever was the diagnosis in those days, today it is typhoid fever .
From Johan Herman's deathbed we have the last glimpses of his irony . A
friend visiting him noticed a big jug of water : "Is it really you drinking
water?" "Yes," the poet replied, "if this is going to be the end, I mus t
reconcile myself to my foe . "
It is likely that Johan Herman left this world with a smile, even if tha t
was gallows humour . His life was in ruins : a bottomless debt, great marital
difficulties and progressive alcoholism . But worst of all : his literary vein ha d
dried up .
He was full of days, but still his friends missed him very much . The Danish
poet Baggesen wrote the last and the lasting epigram for his friend :
Gråt smelted hen i smil, når Wessels lune bø d
Og glædens smil forsvant i tårer ved hans død
It is close to an offence, but I have tried a translation :
Wessel's wit melted weeping into laughte r
He left and smile became tears thereafter
Litteratur e
Follobygderne gjennom prestebriller, Follominne 1969 .
Oslo katedralskole gjennom 800 år, Oslo 1953 .
Den smilende makten . Norske Selskab i Kobenhavn og Johan Herman Wessel,
Bakken, Asbjørn,
Berggrav, Jan,
Bliksrud, L .,
Aschehoug, Oslo 1999 .
Innledning til 'Samlede Digte af JHW', København 1832 .
Innledning til 'JHW's Digte', Kristiania & København 1918 .
Brun, Viggo, Etterlatte notater om Caspar Wessel og hans familie, unpublished .
Christie, Sigrid og Håkon, Norges kirker, Oslo 1969 .
Boye, A .F.,
Bull, Francis,
28
B . Linnestad
MfM
Dunker, Conradine, Gamle Dage, Kristiania 1909 .
Høigård, Einar, Oslo Katedralskoles historie, Oslo 1942 .
Jørgensen, A .D ., Illustreret Tidende, København 27 .12 1885 .
Langberg, H ., Den store Satire, København 1973 .
Levin, I ., Innledning til 'JHW's samlede Digte', København 1862 .
Lie, Jonas, Innledning til 'Udvalgte Digte af JHW', København 1870 .
Lindbæk, Sofie, Fra Det Norske Selskabs Kreds, Kristiania 1913 .
Linnestad, B ., Vestby og skotene, Drøbak 1988 .
Linnestad, B ., Vesen gjennom vidd og vers. Om Jahan Herman Wessel, Oslo 1992 .
Martinsen, Osvald, Vestby Bygdebok I, Vestby 1974 .
Molbech, C ., Dansk poetisk Anthologi I, København 1830 .
Møller, P.J ., 'JHW', fire artikler i 'Figaro', København 31 .1-6 .3 1842 .
Nørgaard, F ., Det norske Selskab og Wessel . 'Frem', København 1 .8 .192 8
Pram, C ., Baggesen, J ., Monrad, P.J ., JHW's samtlige Skrivter, København 1787 .
Schulerud, Mentz, Haflund Gods, Oslo 1974 .
Schwarz, F ., Lomme Bog for Skuespilyndere, København 178 5
Sommerfeldt, W .P (editor), JHW og Norge, Oslo 1942 .
Wessel, Borgmester H .C.H., Nedtegnelser, unpublished .
Wilse, J .N ., Rejse Iakttagelser, København 1791 .
Winsnes, A .H ., Det norske Selskab 1772-1818, Kristiania 1924 .
Samleren, København 10 .6 .1787 .
Nyeste Skilderie af Kjøbenhavn, København 9 .11 1816 .
Nyt Aftenblad, København 5 .8 182 6
Allehaande, København 21 .1 1831 .
Kjøbenhavns-Posten, København 6 .9 1832 & 8 .9 1832 .
Tillæg til Wessels Skrivter, Iste og 2den Udgave .
Morsomme Historier og Anekdoter af og om JHW, København 1879 .
Madame Juuls Stambog, Oslo 1931 .
Norske Selskabs vers-protokol, Oslo 1935 .
46 : 2
MfM 46 :2
29
Thomas Bugge's Journal of a Voyage throug h
Germany, Holland and England, 1777 .
Kurt Moller Pedersen *
1 Travels to England in the 18th Centur y
England had a high reputation among scientists in the 18th Century . This
was not least due to Sir Isaac Newton's scientific achievements that wer e
known and admired everywhere on the Continent . This, however, is not
to say that they were accepted . In France, Cartesian doctrines of scienc e
still flourished though more and more of the younger scientists were take n
by the Newtonian system, e .g . Maupertuis and Clairaut . The results fro m
their scientific expedition to Lapland were taken as a proof of the Newtonia n
world view and scientific notions . It was not only scientists who held a hig h
esteem for England . Voltaire's being exiled to England from 1726 to 172 9
made him an ardent admirer of all English science and the way scientifi c
enterprise was organized . This became clear from his writings, Lettres su r
les Anglais (1734), and Eléments de la philosophie de Newton (1738) .
Scientific expeditions and travels by scientists were interrupted by the Seve n
Years War (1756-1763) between England and France and their respectiv e
allies Prussia and Austria . When the war ended, there seems to have bee n
quite an interest in visiting England, particularly London, Greenwich, Ox ford, and Cambridge . England had been closed for years, and it now becam e
à la mode to travel there . The visitors were all well received by the English . '
It was neither the defeat of the Cartesian world view and its replacemen t
* History of Science Department, University of Aarhus, Ny Munkegade bygn . 521, DK 8000 Århus .
' Lalande 1980, 12 .
30
K .M. Pedersen
MfM 46 : 2
by the Newtonian one, nor was it Voltaire's writings alone that made Lon don attractive . England had a reputation for producing excellent scientific '
instruments . Maurice Daumas gave reasons for this : 2
"The English instrument makers benefited from abundant basic materials such as steel, brass, copper and tin, of consistently high qualit y
and for certain work the English were for a time unsurpassed . John
Bird was the best workman of his period for dividing scales on th e
limbs of instruments, and the most expert at constructing astronomical quadrants ; instruments from his workshop had a universally high
reputation, and for almost half a century were used in all observatories ,
practically to the exclusion of instruments by other makers . "
It was of the uttermost importance for astronomers on the Continent t o
follow all improvements in scientific instrumentation .
Jérôme de Lalande was the first scientist who went there after the war ha d
ended, from March 15 to June 10, 1763 . Jean Bernoulli was there from th e
beginning of December 1768 until late April 1769 . Thomas Bugge arrived in
England on September 7 and left on November 10, 1777 . There were many
other visitors to England at that time, but I have chosen to mention thes e
three travellers because they were all astronomers, and they kept a diary . 3
Jean Bernoulli was the only one of them to have his diary published, bu t
not in an edited version . He had it printed just as it was kept . Lalande' s
diary was published many years later, first in an English translation 4 in
1923, and then in a French version 5 in 1980 . Bugge's diary was publishe d
in a preliminary edition with an English translation 6 in 1997 .
2 Lalande's travels in England, 176 3
Lalande, Bernoulli, and Bugge visited almost the same places in England ,
but the style and content of their diaries are very different . For Lalande ,
2 Daumas 1972, 92 .
3 A Swedish astronomer, Bengt Ferner, travelled through Europe during the Seve n
Years War . He was also in London, and his diary contains several remarks on instrument s
in different observatories and collections of physical apparatus . See Ferner 1956 .
4 Green 1926 .
5 Lalande 1980 . A preliminary version is in Monod-Cassidy 1967 .
6 Bugge 1997 . A brief summary with extract including some of Bugges's drawings fro m
the diary was published in Pedersen 1982 .
MfM 46 :2
Thomas Bugge 's Diary
31
the main objective was to re-establish connections with British scientist s
and scholars . His diary is therefore mainly a list of people he had met .
Personal contacts were the most important part of his journey . There ar e
many descriptions of dinner parties that he enjoyed with colleagues an d
other important people . Let me give a brief example of the style and content
of Lalande's diary :
At the museum, with Lord IVlaclesfield, at the Royal Society, dinne r
at the Mitre Tavern 7 close to Temple with Lord Maclesfield, the ambassador of Venice, Lord Willoughbi, the bishop of Clarence, Short ,
Morton, Birch, Ellicot, Watson, Maskelyne, etc . Ordinary, [the dinner ]
costs three shillings and a sol, but this time, because of the claret, that
is the wine from Bordeaux, which costs five shillings a bottle, we ha d
to pay four and a half shillings . Plum pudding, marque potinger, etc .
Lalande also wrote down small talk about other scientists : "Bradley wa s
tough, jealous, avaricious, melancholic according to doctor Bevis ." We do
not find any professionel descriptions of observatories, instruments, an d
other scientific institutions, as we do in Bugge's diary .
3 Bernoulli's travels in England, 1768-6 9
Jean Bernoulli visited several observatories in Europe, and provides detail s
of the many instruments he saw in his Lettres astronomique . He also re ported about the research projects taking place at the various observatorie s
by interwieving so to speak directors and scientists . In the fall of 1768 h e
was in Göttingen, Kassel, Frankfurt am Main, Mannheim, and from th e
beginning of December he was in London . During his visits to European observatories it became more and more obvious to Bernoulli that many of th e
fine instruments he saw came from London instrument makers . So it must
have been with great expectations that he arrived in the English capital ,
and he was most certainly not disappointed . Not many scientists have bee n
more outspoken in their enthusiasm for their trade than Bernoulli when h e
came to London : 8
7 Later, Bugge also had dinner with members of the Royal Society. See Bugge 1997 ,
"September second I was introduced into the Royal Society Club in Mitre Tavern b y
Dr . Solander . The Society has dinner every Thursday, whereas a non-member is allowe d
to come only every fortnight . "
8 Bernoulli 1771, 63-64 : "je vous ferai part de la surprise agréable où est jeté un Astronome en parcourant les rues de cette Capitale . Vous avés sûrement ouï parler de la
191 :
32
K .M. Pedersen
MfM 46 : 2
I am pleased to share with you the agreeable surprise which meet s
an astronomer when he walks in the streets of this capital . You hav e
certainly heard about the richness and vividness of the boutiques in
London, but I doubt you would imagine how much astronomy con tributes to the beauty of this spectacle . London has a great numbe r
of opticians, and the shops of these artists are filled with telescopes ,
octants, and all these instruments are well-kept, and they flatter th e
eye just as much as the reflections they produce .
Bernoulli does not give any detailed descriptions of the instruments, bu t
he does evaluate the instruments and their makers . We are told that th e
young Dollond is not so well versed in mathematics as his father, though h e
maintains a high reputation for his telescopes with acromatic lenses . Whe n
it comes to dividing the arcs, he is helped by his brother-in-law, Mr . Rams den, "who is considered one of the best instrument makers in London ." 9
Also we learn that "Mr . Nairne is known as a man who produces very goo d
telescopes and other instruments ." 10 Bernoulli spent three days at Greenwich giving a rather detailed description of the buildings and instruments .
He was shown around by Mr . Baily, assistant to the astronomer royal, Dr .
Maskelyne . From there we find the few drawings he included in his diary .
He also visited Oxford and Cambridge : 11 "Literature is primarily studied a t
Oxford, and looking for geometers, astronomers, physicists, &c . one must go
to Cambridge, and there one can convince oneself that science is considere d
to be of importance for the young English student's studies ." Bernoulli is
in no doubt who is the best instrument maker in England : 1 2
The skilled people who are most distinguished for their exactness i n
dividing instruments are Bird, Sisson, and Ramsden , 13 but although
richesse & de l'éclat des boutiques de Londres, mais je doute que vous vous représentiés
combien l ' Astronomie contribue à la beauté du spectacle : Londres a un grand nombr e
d'Opticiens ; les IVlagazins de ces artistes sont remplis de Télescopes, de Lunettes, d'Octan s
&c . Tous ces instrumens, rangés & tenus proprement, flattent l'oeil autant qu'ils imposen t
par les réflexions auxquelles ils donnent lieu . "
Bernoulli 1771, 69 .
' °Ibid . 70 .
" Ibid, 115-116 .
'2 Ibid . 126:
"Les habiles gens qui se distinguent le plus pour l'exactitude des instrumen s
à division sont Mrs . Bird, Sisson & Ramsden, mais quoique ces trois rivaux se disputen t
la palme, je crois, en vous les nommant, leur avoir en même tems assigné le rang dans l e
quel ils sont mis par le plus grand nombre de ceux qui connoissent leurs ouvrages . "
'3 The imstrument makers mentioned are
: John Bird (1709-76), Jeremiah Sisson (1720?-
MfM 46 :2
Thomas Bugge's Diary
33
the three rivals contest the palm ; I think, having mentioned them to
you, that at the same time I have allocated their rank according t o
those who best know their work .
4 Bugge's travels in England, 177 7
" Odense poorly built . . . Assens is a poor town . . . Rendsburg and Slesvi g
are very beautiful towns ." The new director of the observatory in Copenhagen, Thomas Bugge, described in such simple terms some of the citie s
and towns he passed through when in 1777 he set out for a study tou r
to Germany, Holland, and England . Bugge kept a diary and it is full o f
descriptions of all kinds of episodes, incidents, and experiences, written in
brief statements and with a lot of drawings . He left Copenhagen on August
2nd and his first longish stop was in Hamburg, where Bugge observed : 1 4
In the evenings the citizens are occupying the ramparts . Everywhere
in Hamburg luxuriousness of clothing, eating, and game-playing ha s
increased enormously . Moreover, there is little politeness and kindnes s
to foreigners, and these are constantly pestered by hucksters, tailors ,
and Jews, all of them waiting to gain a profit .
Bugge also went to the theater, which he described as follows : 1 5
The playhouse is situated in a miserable corner and has only one exit .
The indoor decorations are tasteless . This evening's performance wa s
zu gut ist zu gut, translated from the English . The director Schroder
played very well . And a certain Brukman played the part of Lofstea d
Bille fairly well . The females were rather poor . The ballet was Vaux hall ; the representation was good ; but the composition and the dancin g
were poor ; the best dancers were the director's wife and a Frenchman .
At that time also young people sometimes got a little crazy : 1 6
The last evening I went to the Vauxhall . It is a small garden whic h
is fairly well illuminated . There were music and singing . The audienc e
was not numerous, but beautiful . The entertainment was poor . This
evening some young Englishmen from the Commercial School behave d
rather badly, pushing other people etc .
83), and Jesse Ramsden (1735-1800) .
14 Bugge 1997 ; 13 .
15 Bugge 1997, 9 .
16 Bugge 1997, 13 .
34
K.M. Pedersen
MfM 46 : 2
From Hamburg he continued via Bremen, Oldenburg, Leer, Nienhrants ,
Windschoten, Groningen, Lemmer, Amsterdam, Leiden, Den Haag, Delft ,
Rotterdam to London where he arrived on September 7th, 1777 .
One of the first institutions he visited was the Foundling Hospital, established in 1739 "for the maintenance and education of exposed and deserte d
young children . . . to prevent the frequent murders of poor miserable children at their birth, and to suppress the inhuman custom of exposing new born infants to perish in the streets ." 17 When Bugge visited the Hospital ,
we are told there were "120 of each sex . It was said that about the sam e
numbers were in the country for their health ." 18 The children's state of mind
could be improved, so it was thought, by exposing them to the more obscur e
sides of social life, and thus giving the children a dislike for the vulgar life .
Whether this was the official strategy or not I do not know, only that th e
children every day had a view of a characteristic picture by Hogarth : 1 9
In a western room is a painting by Hogarth ; it is called Hogarth's Master Piece . It represents the English Guard Regiment's march agains t
the Scotch rebels . Two women, one with a child and another who i s
pregnant quarrel over a soldier . One soldier is drunk and falls on the
ground ; one of his companions wants to give him some water, but h e
reaches for a glass of aquavit which a sales apprentice offers him . Another soldier overturns a girl's milk pail and lets it run into his hat . A
chimney sweeper's boy holds his black skullcap underneath . Another
soldier fingers a girl underneath her skirts, as she climbs up in orde r
to watch two men boxing .
Thomas Bugge did not set out on his journey to study art and sociology . H e
wanted primarily to study observatories and their instruments so that, afte r
his return, he could initiate a restoration of the observatory at the Roun d
Tower in Copenhagen . The expectations of the outcome of his journey wer e
great . Danish astronomy should again be brought back into the front line, a s
the heritage from Tycho, Longomontanus, Romer, and Horrebow demanded .
These demands to Bugge were not rooted in his theological degree fro m
the university of Copenhagen, but in his background as a mathematician ,
17
The Encycl .opædia Britannica, Thirteenth Edition, 1926, volume 9, 747 . Entry :
Foundling Hospital .
18 Bugge 1997, 119 .
19 Bugge 1997, 121 . I have found a print resembling this description, see Fig . 1 .
MfM 46 :2
Thomas Bugge '3 Diary
35
Figure 1 : In 1745 a regular and volunteer force encamped at Finchley, 7 miles fro m
St . Paul's Cathedral to resist the Pretender . The gathering of this force inspire d
Hogarth's famous picture "The March to Finchley" . From Engravings by Hogarth ,
ed . Sean Shesgreen, Dover Publications, New York 1973 .
having been supervised by Professor Hee, and since 1759 as an observe r
at the Round Tower . From 1762 to 1777 he was director of the surveyin g
of Denmark . More important, however, was his appointment as professo r
of mathematics, and director of the observatory in 1777, the year when h e
travelled to England .
36
K.M. Pedersen
MfM 46 : 2
The new director immediately saw that the observatory had to be modernized and better equipped . It is quite interesting to observe that almost al l
the new instruments that Bugge procured after his return from Englan d
were either bought in England or were constructed by the local instrumen t
maker, Johannes Ahl, in accordance with Bugge's instructions which originated in what he saw in England . In what follows I will show how some o f
the new instruments and devices at the Round Tower can be traced bac k
to drawings and descriptions found in Bugge's diary .
The diary has 184 pages and is full of drawings of the many instrument s
he studied . It is a very important historical document allowing historians o f
astronomy to register instruments that later disappeared . It is not only the
drawings that are of importance . Bugge's descriptions of what he studied
also allow a better understanding of instruments still in existence in Britis h
museums . He visited the observatories in London, Oxford, and Cambridge .
Just as important, however, were the contacts he established with leadin g
English astronomers, and he was introduced to the Royal Society and th e
Royal Agricultural Society . In what follows I shall concentrate on such descriptions and drawings as pertain to instruments and devices later to b e
found in the observatory at the Round Tower in Copenhagen .
5 Illumination of wires in transit instrument s
On September 23rd Bugge visited Richmond, about 14 kilometers south west of London where the king had his summer residence . In a corner of the
park was an observatory : 2 D
I went to Richmond where I got the opportunity to make the acquain tance of Doctor de 1Vlembray 21 and his son-in-law Mr . Rigaud . They
were both kind enough to take me up to the observatory which th e
king has erected for his own pleasur e22 . On the bottom floor or in th e
20 Bugge 1997, 147 .
21 Stephen Demainbray (1710-1782) .
22
King George III had become very interested in the transit of Venus in 1769 and buil t
his own private observatory at Richmond (now Kew Observatory) where he could observ e
the transit himself . The observatory was designed by Sir William Chambers and wa s
ready by the time of transit in June 1769 . Stephen Demainbray was the observatory' s
first superintendent . When he died, he was succeeded by his son, also called Stephe n
Demainbray . See Morton and Wess 1993, 29 and 117 .
MfM 46 :2
37
6
Figure 2 : The transit instrument at the observatory in Richmond .
basement there are mathematical workshops . On the first floor there
are several rooms . a) one housing the transit instrument ; the telescop e
is 5 feet, the axis 3 å . At the top of the solid block A was a devic e
for the lantern [see Fig . 2] . a and b are two stationary brass stick s
with holes, through which the triangle cde can be moved . At the end
F the lantern illuminating the filaments is placed . At the other end i s
placed a bar cg with a counter-weight, so that the lantern remains i n
the wanted position.
Bugge gives us here a description of a transit instrument that can mov e
around a horizontal axis in the north-south meridian . Inside the telescope i s
a thin horizontal and several vertical wires, the middle of which determine s
the meridian . These wires serve to determine the coordinates of the stars .
The method requires that the wires can be seen, and therefore they must b e
illuminated by a lamp, and this is what interested Bugge . He carefully tell s
how this is done at the observatory in Richmond . Whenever the telescope i s
rotated the lamp must follow and that mechanism Bugge carefully drew i n
his diary . But how does the light from the lamp enter the telescope? If the
light from the lamp should enter the telescope properly, the lamp should b e
placed in front of the telescope, in which case the stars could no longer be
38
K.M. Pederse n
MfM 46 : 2
Figure 3 : A transit instrument in Oxford .
seen . What they did at Richmond, as can be seen from Fig . 2, was to plac e
the lamp close to the axis of the telescope so that sufficient light entered th e
telescope . That was, indeed, a rather simple solution to the problem of illuminating the wires . In general, the collection of instruments at Richmon d
was not of the highest quality ; they were made by "many people, some o f
whom were not highly skilled ." 23
However, in Oxford Bugge found another, more ingenious device for illuminating the wires : 24
A is the axis of the transit instrument [see Fig . 3] . ab is a circle o f
mahogany around the axis attached to the pier without being connected to the axis . Round it is another circle cdef , connected to the
piece hg which is of about the same length as the telescope . Q is th e
lamp, illuminating the circular piece of polished brass MN with th e
requisite aperture O . At P a. weight counterbalances the lamp . Whe n
the telescope has been adjusted to focus upon the star, the observer ,
23 Morton and Wess 1993, 123-24 .
24 Bugge 1997, 225 .
MfM 46 :2
39
Figure 4 : Ahl's transit instrument made for the Round Tower in Copenhagen . Fro m
Thomas Bugge : Observationes astronomicæ annis 1781, 1782 & 1783 . Hauniæ 1784 .
with one hand, moves the piece hg and the lamp Q up and down until
he finds that the wires are well illuminated .
It is easier to see how the device worked from a drawing of the transi t
instrument that Ahl made at Bugge's request and in accordance with hi s
instructions [see Fig . 4] . The lamp is suspended from a rod that can mov e
around the horizontal axis of the telescope . Bugge did not use the rathe r
complicated suspension he saw while at Richmond, but the simpler device
K.M. Pedersen
40
MfM 46 : 2
Figure 5 : A transit instrument from the Geodetic Department in Copenhagen ; no w
in the Steno Museum, Aarhus . Photo : Hanne Teglhus .
from Oxford . The light from the lantern is reflected in a brass mirror tha t
has a hole in the middle such that the light from the star and the lanter n
can enter the telescope . The same device illuminating the wires is found on
a transit instrument from the Geodetic Department in Copenhagen, no w
at the Steno Museum in Aarhus shown in Fig . 5 . From the drawing o f
Ahl's instrument we can see that two counterweights are suspended at each
extremity of the horizontal axis of the telescope, a device Bugge also saw a t
Oxford : 2 5
25 Bugge
1997, 221 .
Mf1VI
46 :2
41
p
'11 M•1
n _ n M w~iÿn
~
..
.:i
.i
1 ff~• ~. ./•.~ ~
Y ~.•
f.
.. :MMa :••
::
! : :. :
:
'
..
. .o .1
. . .i a
II •w
M
.Ii/å01h .i•n . ~
• 10
iY• .~~ MO C S:i /
1 miss
==
• n ft:1.M.t.I.. C
:
i-
Figure 6 : A transit instrument with counterweights from Oxford .
The whole weight of the instrument was more than 100 pounds, an d
in order to relieve the weight from the pivots, levers (bed) are placed
at its extremities [see Fig . 6] . Their centres of suspension c rest o n
the iron bars AB . bd and ba are made of mahogany . The latter (ba )
is shown in figure 2 [here Fig . 6] . The weight p counterbalances hal f
of the weight . With one finger I could lift the instrument out of it s
bearings at a.
The director of the observatory at Oxford, Dr . Hornsby 26 , was proud t o
guide Bugge through the buildings . The observatory was new and modern ,
in fact it was only completed the year after Bugge's visit . Hornsby had aske d
the trustees for astronomical instruments "to be made by the best instrument maker of the time, John Bird" 27 , and it was accepted . The observator y
was "one of the best equipped of its time ." 28 Bugge made a lot of notes an d
drawings in his diary . Immediately after his return to Copenhagen Bugge
began reconstructing the observatory, and it was completed in 1780 . 29 It i s
difficult to relate details in the new construction to particular construction s
26 Thomas Hornsby (1733-1810) is remembered for his active part in the foundation o f
the Radcliffe Observatory in Oxford of which he became the first director . The buildings
were completed in 1778 .
27 North 1972 . That Bird was the best instrument maker is a statement in accordanc e
with Bernoulli's observation as I quoted in section 3 . The observatory's mural quadran t
was one of the best instruments produced by Bird, see Bernoulli 1771, 116 .
28 Daumas 1972, 124 .
25 Bugge 1784, xxvii and Tab . I . A reconstruction of Bugge's observatory is i n
Gyldenkerne & Darnell 1990, vol . 2, 214-215 .
42
K.M. Pedersen
MfM 46 : 2
in England, but we can see from Ahl's instrument (Fig . 4) that the axis wa s
counterbalanced in the same way as in Oxford (See Fig . 6) . We do not kno w
anything about the instrument shown on Fig . 5, but it might well be that i t
is an instrument made at Bugge's request, since it has a circular, reflectin g
ring in front of the telescope, as described by Bugge and shown in Fig . 3 .
Moreover, the spirit level is suspended from the axis in a way similar to th e
ones shown in Figs . 6 and 7 . Bugge was very impressed by the observator y
in Oxford and its equipment : 30
Not without regret did I leave Oxford Observatory which is no doub t
the best in Europe, both as regards the arrangement and instruments .
Professor Hornsby most courteously assured me of his friendship an d
correspondance .
6 A spirit leve l
Already in September, Bugge visited watchmaker Cumming 31 where he saw
a spirit level : 32
When the spirit level ab [see Fig . 7] had been adjusted by proper rever sal in the horizontal position of the meridian circle cd, Mr . Cummin g
noticed that the bubble did not remain in the centre when the tele scope was elevated to 20, 30, 40, etc . From this he concluded that the
axis was not on the same level or on a level parallel with the length
of the spirit level . In order to change this he had fitted a screw gh, so
that the entire spirit level can be moved to both sides .
The adjustment of this screw must then be continued until the bubble
remains at its marks in all positions of the meridian circle .
Before Bugge went to Oxford he had paid John Russe11 33 a visit and he saw
him again after he had returned from Oxford . Bugge noticed a spirit leve l
in his possession and made a drawing of it and described it as follows : 3 4
30 Bugge 1997, 261 .
'Alexander Cumming (1733-1814) .
32 Bugge 1997, 169 .
33 John Russell (1745-1806) was a famous painter who was also interested in astronomy.
He drew an exceedingly accurate map of the moon . Bugge's description of many of Russell's astronomical instruments is of great importance, since they had never before bee n
recorded .
34Bugge 1997, 275 .
MfM 46 :2
Thomas Brigge 's Diary
43
Figure 7 : A transit instrument with spirit level as seen in watchmaker Cumming' s
shop in London .
AliiùÎ4(
f8W11 Il111illl1Rl
Figure 8 : Ahl's spirit level made in Copenhagen . From Thomas Bugge : Observationes astronomicæ annis 1781, 1782 & 1783, Hauniæ 1784 .
He [Russell] also had an excellent spirit level, 6 inches long, made b y
Bird, and constructed in the same way as the Centre or Trial Telescop e
described by Smith in his Optics . He places the spirit level on th e
quadrangular plates as shown in the following figure [see Fig . 9] .
In Copenhagen Bugge had Ahl construct a spirit level in accordance wit h
the ideas he had learned about in England, as can be seen from the drawing
44
K.M. Pederse n
MfM 46 : 2
of Ahl's instrument [see Fig . 8] .
7 Conclusio n
It is clear from the diary that Bugge learned much about observatories an d
the new equipment in their possession . He brought back new ideas that
eventually were used in constructing new instruments in Copenhagen . It is ,
however, important to make clear that Bugge never went into details whe n
it concerned precision instruments and their working . Perhaps, he was no t
told about them . It was kept as a secret how the more subtle details wer e
produced and manufactured, as Dan C . Christensen made it clear in his paper about spying on scientific instruments . 35 Furthermore, as Christense n
has also pointed out, it would have been impossible to construct precisio n
instruments in Copenhagen because of lack of knowledge and skill . After
Bugge's return, Bidstrup was sent to London to learn the trade, and hi s
journey to and stay in England were recommended by Bugge . From his stay
in England Bugge had learned, I believe, the importance of having skille d
instrument makers in Copenhagen, if the observatory was again to be recognized as a leading one in Europe . The observatory did not obtain that
position while Bugge was its director . He was not able to supply the fund s
for it, and Bidstrup died before he could begin his trade . Bugge's diary is a
much more valuable document than those of Lalande and Bernoulli . It contains a more detailed description of how the observatories were equipped .
Today his diary is recognized as a major source of English astronomy an d
scientific instrumentation . On the 300th anniversary of Greenwich Observatory in 1975 Bugge's diary returned to England to be exhibited for som e
weeks in Greenwich where it attracted much attention from the hundred s
of historians of science and astronomy gathered to celebrate this important
event .
Acknowledgment
Bugge's diary is in The Royal Library in Copenhagen (Ny kgl . Saml . 377 7
e4o) and I am indebted to the Library for permission to reproduce figs .
2,3,6,7 and 9 .
35 Christensen 1993 and 1999 .
MfM 46 :2
45
.4 A
Figure 9 : Bugge's drawing of Greenwich Observatory, October 1777 . Text : Facaden
at Greenwich Observatorium, saadan som det sees fra Hospitalet . Var jeg hos Mr.
Russel, som viiste mig end viidere adskellige af sine Instrumenter. Et watterpas ,
sent fra en af hans venner, hvilket til smaae operationer var scErdeles vel skikket.
Translation of the text : The front of Greenwich Observatory, seen from the Hospital .
I visited Mr . Russel who showed me several of his instruments . A spirit level sen t
from one of his friends . It is very suitable for smaller operations .
46
K.M. Pedersen
MfM 46 : 2
References
Bernoulli, Jean, 1771, Lettres astronomiques oö l'on donne une idée de l'état actuel d e
ll'astronomie pratique dans plusieurs villes de l'Europe . Berlin : Chez l'auteur .
Bugge, Thomas, 1784, Observationes astronomicæ annis 1781, 1782 E4 1783 . Hauniæ :
Nicolai Möller .
Bugge, Thomas, 1997, Journal of a Voyage through Holland and England, 1777 . Edited
by Kurt Møller Pedersen . Aarhus : History of Science Department .
Christensen, Dan Ch ., 1993, Spying on Scientific Instruments . The Career of Jesper Bidstrup . Centaurus, 36 209-244 .
Christensen, Dan Ch ., 1999, English Treatment Makers Observed by Predatory Danes ,
this volume, 47-6 3
Daumas, Maurice, 1972, Scientific Instruments of the 17th and 18th Centuries and thei r
Makers . London : Portman Books .
Ferner, Bengt, 1956, Resa i Europa . En astronom, industrispion och teaterhabitué geno m
Danmark, Tyskland, Holland, England, Prankrike och Italien. 1758-1762 . Utgive n
med indledning och register av Sten G . Lindberg . Lychnos bibliotek nr . 14 . Uppsala :
Almqvist & Wiksells boktryckeri .
Green, F .C ., 1926, The journal of Lalande's visit to England . The History teacher 's miscellany . 4 113-118 ; 140-144 .
Gyldenkerne, Kjeld & Per Barner Darnell (eds .), 1990, Dansk astronomi gennem firehundrede år. København : Rhodos .
Lalande, Joseph-Jérôme Lefrançais de, 1980, Journal d'un voyage en Angleterre 1763.
Publié avec une introduction par Hélène Monod-Cassidy. Oxford : The Voltaire
Foundation . The original journal is in Bibliotheque Mazarine, Paris .
Monod-Cassidy, Hélène, 1967, Un astronome-philosophe, Jérôme de Lalande . Studies on
Voltaire and the Eighteenth Century .
Morton, Alan Q . & Jane A . Wess, 1993, Public êl Private Science . The King George III
Collection. Oxford : Oxford University Press .
North, John ; 1972, Hornsby, Thomas . Dictionary of Scientific Biography . New York :
Charles Scribner's Sons . Vol . 6, 511-12 .
Pedersen, Kurt Møller, 1982, Uddrag of Thomas Bugge's dagbog 1777 . Bibliotek for læger.
174,151-164 .
MfM 46 :2
47
English Instrument Maker s
Observed by Predatory Dane s
Dan Ch . Christensen *
The title of this paper is hinting at the fact that for a middle sized monarchy
like Denmark-Norway scientific instruments were a prime requisite for th e
progress of scientific research, i .e . astronomy, experimental physics, geodesy ,
as well as navigation . For shortage of domestic producers the obvious re source to turn to would be the instrument makers in London renowned fo r
their leading position . In theory, there were four methods of transferrin g
technology :
1.
buying instruments abroad ,
2.
enticing skilled artisans to emigrate and work in Denmark,
3.
spying on instrument makers abroad, an d
4.
sending talented artisans abroad for training .
The first method simply consisted in buying instruments abroad . Professional astronomers like Thomas Bugge r or amateur natural philosophers like
A .W . Hauch 2 and the Reventlow brothers 3 went to the boutiques of Ramsden, Nairne & Blunt, and Adams to buy the instruments they adored . Thes e
Danes were on their grand tours or study tours and usually they were law abiding travellers . But purchasing instruments was a simple activity tha t
meant little for promoting a national production .
* Roskilde Universitetscenter, Postbox 260, DK-4000 Roskild e
1 Thomas Bugge, Journal of a Voyage through Holland and England, 1777 . Ed . by Kur t
Møller Pedersen, History of Science department, University of Aarhus, 1997 . See also Kur t
Møller Pedersen's paper in this volume .
2 Hemming Andersen, En videnskabsmand af rang, A . W. Hauch, 1755-1838, Århus .
3 Unpublished travel diary by C .D .F . Reventlow, 1769-1770, The Reventlow Museum ,
Pederstrup, Denmark .
48
D. C. Christensen
MfM 46 : 2
Secondly, the government might send out agents to try to entice skille d
workers to emigrate to Copenhagen . In fact, the only instrument maker o f
competence in Denmark before 1800 was a Swede, Johann Ah l 4 , the journey man of Daniel Ekström, Stockholm, who left Sweden possibly to rid himsel f
of a personal debt . Bugge, who had seen the most advanced instrument s
on his journey to Paris and London during 1777, had Ahl imitate craft y
innovations in his Copenhagen workshop on the basis of his sketches . 5
This strategy, however, if employed in England, would violate the tool act s
which illegitimated the enticement of skilled artisans to go abroad . Th e
objective, of course, was to maintain a leading position on the world market .
There are many examples of British artisans entering into Danish-Norwegia n
service and setting up workshops in the twin-monarchy with governmen t
support . 6 The glass, machine tool, iron foundry, and agricultural implemen t
industries are typical examples, but to my knowledge no British instrumen t
makers immigrated here .
The third strategy was industrial espionage . The objective of the spy
would be to make descriptions and drawings or to get hold of models o r
prototypes of advanced tools or machines from leading manufacturers . Onc e
this had been achieved, however, another problem arose : how to smuggl e
the goods out of Britain? Spies belong to the historian's most appreciate d
agents . Not only because the historian shares everybody's fascination with
spy stories, but because the spy is the eminent supplier of reliable evidenc e
for the simple reason that in order for his intelligence to be useful for plagia rism it had to be thoroughly understood . Hence his notebooks are usuall y
superior in quality to the information supplied by manufacturers who too k
an interest in shrouding their trade in mystery.
In connection with espionage it deserves mentioning that the tool acts als o
illegitimated the exportation of core machinery of production . ? The objec4 See Olov Amelin's paper in this anthology.
5 See Kurt Møller Pedersen's paper in this anthology.
6 This is elaborated in Dan Ch . Christensen, Det Moderne Projekt . Teknik og Kultur i
Danmark-Norge 1750-(1814)-1850, Copenhagen, 1996 with an English Summary . List o f
spies, pp . 507-508 .
' A series of acts starting 1719 to prohibit the exportation to foreign parts, of tool s
and utensils used by the iron and steel manufacturers of Great Britain and to prevent th e
seducing of artificers or workmen, employed in those manufactures, to go into parts beyon d
the seas . 25 . Geo . III cap . lxvii, pp . 286-291 . See John R . Harris, Industrial Espionag e
and Technology Transfer : Britain and France in the 18th Century, Variorum, Hampshire,
MfM 46 :2
Industrial Espionage in England
49
tive was to protect outstanding inventions from falling into foreign hands ,
just as patent bills protected the inventor from being exploited by domes tic plagiarism . It should furthermore be noticed that British manufacturer s
were divided according to opposed interests . As we shall see some manufacturers like for instance Wedgwood would put pressure on Parliamen t
to extend and tighten the tool acts, whereas a group of members of Th e
Society for the Encouragement of Science, Agriculture, &Commerce, London, demonstrated their idealistic belief in a universal brotherhood of ma n
by publishing patent drawings and descriptions in an international journa l
titled The Repertory of Arts, Manufactures, and Agriculture . The paradox was that in order to have a patent legally recognised a description an d
sometimes a drawing of the patented artefact had to be submitted to th e
court ; this procedure made the patent public knowledge . Since the patent
was valid in Britain, but not abroad, there was no legal obstacle to us e
this knowledge to plagiarise the innovation abroad . An illustrative example
would be Bryan Donkin's paper-making machine from 1808 . An issue of Th e
Repertory containing patent drawings of this machine appeared in Denmar k
and a skilled artisan, Ole Winstrup, made efforts to imitate it . 8
There were a number of Danish-Norwegian civil servants spying on Britis h
manufacturers between 1760-1807 . 15 spies have been identified, many o f
them professors and free masons . 9
The most eminent Scandinavian spy, no doubt, was Professor J .M . Ljungberg, a Swede by birth, operating on behalf of the Danish government an d
Count Schimmelmann, Chancellor of the Exchequer and industrialist in a
wide range of manufacture . He had several sojourns in England in the las t
quarter of the eighteenth century procuring intelligence from a good man y
inventors behind the Industrial Revolution, e .g . Boulton, Watt, Chippendale, Wilkinson, Arkwright, Bramah, Wedgwood, Garbett, Smeaton, Rams den, and Troughton . In fact, during 1788-89 he stayed for more than a yea r
in Birmingham getting well acquainted with Boulton, who employed him a s
an agent to sell furnishings to Chippendale and issued letters of recommendation to manufacturers using his tools and steam engines, like Wedgwood
and Wilkinson . The intelligence he collected was meticulously taken dow n
1998 .
8 The Repertory of Arts, Manufactures, ti Agriculture, vol . 13, 2nd series, Pl .ix, London
1808 . The Danish attempt is described in DMP (see note 6), p . 460-462 .
9 cf. note 6 .
50
D . C. Christensen
MfM 46 : 2
in his notebooks 10 and he acquired a large quantity of tools (a consignmen t
of four boxes weighing almost half a ton), including some `clock maker' s
tools', which was a euphemism for precision tools for instrument makin g
that were blacklisted by the tool acts .
Now, just as the making of scientific instruments demands its skills and experience, so does the noble business of espionage . And just as Ljungberg kne w
that industrial secrecy made spying a high-risk enterprise which challenge d
the ingenuity of the predacious agent, so he was himself utterly reluctan t
to disclose his methods . His notebooks are extremely scant with information on how he contrived his business . However, his spying on Ramsden' s
tube drawing machine reveals an interesting case . As Ljungberg well knew ,
Ramsden was very secretive . At first he managed to get one of Ramsden' s
workers to open his mouth and took down his oral report as follows :
Instrument makers use brass tubes for telescoping . The tubes must b e
exceedingly accurate in order to be drawn out and pushed in smoothly .
Mr . Ramsden and Mr . Wright have recently drawn patents for plate d
telescopes . This is done in a certain way similar to the drawing of a
wire and the process is as follows :
The brass sheet for the tubes is made twice as thick as it is going t o
be . It is then hammered and soldered with the diameter it is goin g
to retain ; the soldering is brass and spianter (what this is I'm not
fully aware) . One then strikes it on to a steel mandrel which is perfectly cylindrical and indeed very accurately turned and well polished .
(added marginal note : I think that the mandrel is in one piece an d
that the split pin only enters at the neck cc to fix the mandrel) . Thi s
mandrel has a neck in cc into which one beats firmly the edge of th e
tube . Two halves of a certain pair of thongs fit into this neck . They
are pressed together and afterwards they are fixed with two split pins .
The mandrel is part of a drawing machine like the usual ones, and if
I understood it correctly it is operated by means of a chain, simila r
to a clock's chain, i .e . a crank handle and a gear . But this operatio n
is vertical, not horizontal as usually . A steel disc with a round hole
of appropriate size is fixed in the groove, and the brass tube is draw n
' °J .M . Ljungberg's notebook, Kungl . Myntverkets arkiv, Varia I d, Riksarkivet, Stockholm . The notes amount to 273 pages, written in German apart from a few pages i n
English, richly illustrated . The entry on Ramsden's tube drawing machine is from Jun e
1789 .
MfM 46 :2
51
between this steel disc and the mandrel, so that it stretches and be comes smooth and round . This process is repeated several times ; and
usually, at least as far as the plated tubes are concerned, they com e
out twice as long as they entered .
The cost of this machine is £200 .
This intelligence stems from an oral account only, becaus e
the machine cannot be observed. 1 1
Now, one might think that a tube drawing machine is a comparatively simple device . But reconstructing one on the basis of this oral account woul d
hardly be possible, since it contains too much doubt, possible misunder standings, and lacks measurements . Furthermore, it leaves no clue as to th e
construction either of the transmission from handle to chain or of the disc hole through which the tube is drawn . So, upon second thoughts Ljungber g
must have found his account rather flimsy and requested a drawing statin g
all the precise measurements and a detailed knowledge of how to operate i t
and how to tackle its most frequent malfunctions . And here it is :
AAB is a stand of wood in which a groove B is cut out, 5 inches deep ,
5 inches wide . The inside of the upper half is covered by iron rulers a s
on two sides, and where the stand ends, bent in b and as they ascen d
they also widen and serve as a frame, c, for pivots for the wheels ,
CDEF . In order to obtain more strength with less mass, rulers of 3/8"
are forged on the edge upwards vertically and similarly horizontally .
C is a cylinder of wood sitting together with an iron cog wheel, F, on a
shaft having its pivots lying in the stand cc . Around this cylinder roll s
an iron chain, h k 1, which in 1 is bolted very firmly on the cylinde r
and descends freely in the groove edged by the rulers, ab . The form
and property of the chain is apparent in the drawing, and at its lowe r
end is a shackle, h, of strong iron .
D is a large wheel of wood, nearly 3' in diameter, and it serves bot h
as a drive for the transmission and as a flywheel . It has its shaft in
the drive d, which gears into the iron cogwheel, E . On the shaft o f
the wheel E there is also the drive, f, which gears into the wheel F ,
and turns the cylinder C to which it is fixed so that the force on C
gets very strong because the drives are as small as the requirement s
of strength permit .
Fig . 3 shows the disc which is abt . 4 1/2" square and 3/4-1" thick .
One section of the inner ring is conical, nn, whereas the other, mm ,
is cylindrical, and it is this latter section that works on the tube . Th e
11 Ibid ., p . 147, translation and fat italics are mine
.
52
D . C . Christensen
1VIfM 46 : 2
width of section mm actually touching the tube is only small, abt .
1/154/8" . The disc consists of two parts : the square G and the ring
mn,mn . The former is made of iron, only the latter is of steel . Thi s
ring is turned and soldered with brass on the iron G . Then one glows
the whole disc and puts it into water by which the ring mn obtains
the appropriate level . One still needs to polish it afterwards .
Fig . 5 is a steel mandrel that is turned perfectly round and smooth .
At both ends, rs, it is turned thinner and is pierced with two oblon g
holes, p and q . These serve to fix to the shackles tt by putting the pin ,
u, through all three holes . The mandrel H is made of steel and strongly
heated, then turned and well polished . One has as many mandrels a s
one wants to make tubes of different diameters, and so with the discs ,
G . The pin is quite easy to put in and out . The machine is fixed t o
the floor and to the wall to make it stable .
The machine is used as follows :
The brass for the tubes is rolled sheet . One cuts it in strips of appropri ate size and hammers it to the shape of a tube on a long steel cylinder
which is only fixed at one end . When the tube is appropriately larg e
and round it is hard-soldered by brass and spianter which makes the
brass relatively more fluid . After the soldering one hammers it on a
polished cylindrical bar the one end of which one puts into the hol e
of a round vertical wooden beam to fix it . This rod appeared slightly
conical to me, and on this one hammers the tube perfectly round, an d
the soldered places thin and even . The bar is as thick as the inne r
diameter of the tube, 4 to 5' long and while it was beaten it lay, presumably, askew relative to the horizon, abt . 30° . I suppose that he
turns the tubes quite often .
One puts the tube being 1 1/4" thick and 5" long on the steel mandre l
H, so that the edge can be folded 1" or 3/4" wide over the neck, r .
One beats this edge with a wooden hammer to fix the tube firmly. I
think the cylinder was probably tallowed beforehand .
One then took the shackle t and put it through the shackle of the chain ,
h, in the machine, kept the neck H r of the mandrel H in the shackl e
t, so that the holes fitted into one another and inserted the split pin
u . Before this, however, the upper end of the tube had been tallowe d
and a disc like G with a hole slightly narrower than the diameter o f
the tube was put on top of the neck r . Obviously, the chain had bee n
lowered by turning the wheel D so that the neck r was in the lowe r
part of the groove B, where the disc G has sufficient room, becaus e
higher up between the rulers as the distance is smaller than the widt h
12 Ibid ., p . 149 .
MfM 46 :2
R063, :.. 46..."
74-14 2'; ~-~ . .,
,*,wo.,
53
Jc• r.. ~ s'~1 . ...i..~.
>n : .
Ø
ti
r
,i'
L
- ..1--,
,r '~ ,a
'
I
Figure 1 : Ill . Ramsden's tube drawing machine
12
54
D . C. Christensen
MfM 46 : 2
of the disc G .
By turning the chain upwards the mandrel H ascends together wit h
its fixed tube and the disc G . Only the disc eventually hits the section ii between the rulers and is stopped while the mandrel with th e
tube continues to move . In this way the tube is drawn while stretch ing considerably . When it is through one puts the pin out, takes th e
cylinder from the shackle and removes the disc and inserts another
one yet slightly more narrow ; each time one must be careful to tallo w
the upper part of the tube and when the disc is stopped at ii, direc t
the mandrel H in the vertical position manually . In this way the tub e
passes 4 or 5 rings when, finally, it has become twice as long as i t
was in the beginning and so even, smooth and round as if it had bee n
turned on a lathe with the utmost care .
Mr . Ramsden is said to have had this machine for ten years .
Normally, the tube is fixed very firmly to the cylinder or the mandre l
making it quite laborious to remove it . This is done in the followin g
way : One takes the mandrel out, turns it around and puts on it a dis c
that fits the mandrel quite accurately, but does not let the tube sli p
through . One hangs the mandrel on the shackle and turns upward s
again, so that the ring removes the tube, which is not possible withou t
employing considerable force . However, in case the upper end of th e
mandrel rises very high in the machine, so that the space is insufficient ,
a loose wooden frame, three-sided and about 1' long, is available t o
be placed under the section ii in order to extend the narrow groove ;
and the disc G hits this frame like earlier it hit the section ii . Th e
drawing shows the chain too long to enable one to see how it is fixe d
to the cylinder C . Obviously, it is no longer than the periphery of th e
cylinder C . 1 3
Now, throughout the historiography of technology there has been a lot o f
fuss about internalism which has been loathed as `nuts & bolts' histor y
a despicable hobby-activity usually practised by retired engineers . Goo d
history of technology should focus on the context and hence it can be re searched and written by ordinary historians unfamiliar with the intricacie s
of the technological artefact itself . The case of the tube drawing machin e
shows, however, that unless we pay attention to the very function of tech 13 Ibid ., pp . 148-152 .
MfM 46 :2
55
nology it is hard to appreciate fully the task of technology transfer . In othe r
words : an internalist approach is not a superfluous alternative to an externalist approach, but a necessary and indeed complementary asset to th e
contextual history of technology.
The first and the second piece of intelligence collected by Ljungberg ar e
worlds apart, and nobody was more aware of the importance of exactitud e
for successful technology transfer than Ljungberg himself . As far as I know
this drawing and description of a tube drawing machine is the only exist ing piece of evidence illuminating a basic technology of the production o f
scientific instruments left to the historian . Furthermore, the step from th e
oral to the written account marks two stages in espionage methodology . It
must have demanded exquisite gifts of persuasion to entice one of Rams den's workers to reveal the patent of the tube drawing machine . But whe n
we learn how Ramsden treated his workers we begin to understand tha t
Ljungberg must have spent quite a sum to bribe his informant to obtai n
sketch and measurements as well . Here is what he reports from inside th e
walls of Ramsden's workshop . 1 4
The ordinary workers who file, turn, plane, etc . earn from 18 to 21 s .
per week of 6 days . [The foremen earn a lot more according to their
skills] . They work daily 12 hours, from morning at 6 o'clock till 8 .3 0
p .m ., since they have 1 hour from 1 till 2 p .m . for lunch, half an hou r
for breakfast, and out of the 12 hours he gives them a quarter of a n
hour in the afternoon for tea . 1 5
In the workshop there is a slate on which everybody writes down th e
time of his arrival . In case he comes 2 minutes past 6 he will write 6 .1 5
and is paid accordingly. If he writes a quarter of an hour too much h e
is fined 7d ., and for each hour 28d ., or 2s . 4d . There is a clock in th e
shop and the other workers usually control the slate .- If one worker
pushes another he pays 1s ., if he beats him 2s . 6d . To show up drunk
in the shop is fined, and to bring a stranger into the shop costs 2s . 6d . Every Saturday the workers receive their wages with a deduction fo r
the hours they have not been working and for fines, which money i s
given to the workers who will then drink ale or porter for part of it . #
14
Anita McConnell, "From Craft Workshop to Big Business - The London Scientifi c
Instrument Trade's Response to Increasing Demand, 1750-1820", London Journal 19, 1 ,
pp . 36-53 .
15 Here Ljungberg is one hour wrong, ibid
., p . 152 .
D. C. Christensen
56
MfM
46 : 2
If a worker shows up in the shop with dirty hands or without havin g
shaved he will get fined . In other workshops the rules are different .
Ramsden's rules are pinned up in his workshop .
Is this so remarkable that it deserves mentioning? Well, to Ljungberg thes e
labour conditions differed from those he knew in Denmark where the guild
system protected the journeymen and required masters to abide by the rule s
common to all members of the guild . Hence, when he found that individual
instrument makers in London were capable of setting up private rules o f
conduct he was astonished . The secrecy and the costs and risks to circumvent it were another surprise that Ljungberg got used to during his stay i n
Britain . He reported to Schimmelmann that this modern profession of instrument making could not be contained within the framework of the guil d
system . 16 Here in London a new division of labour, a new system of credit, 1 7
a new market of competition and secrecy and a new power-relation betwee n
employer and worker had established the pillars of modern capitalism .
A great improvement by the local instrument makers consists in th e
fact that they make smooth and even work such as rulers, discs of
brass with a plane . The plane is made of cast iron smoothed over below . The iron is adjustable by one screw and can be fixed by another .
This provides for the wide, smooth planes, the straight edges and sharp
corners, and much time is gained . The plane costs 1 1/2 Guinea . Ramsden employs about 40 workers, of whom 9 glass polishers, . . . filers ,
. . . planers . . . , turners, 1 draws the tubes, . . . divide the instruments .
He manufactures almost anything at least as far as telescopes and bi g
instruments are concerned, in his own workshop, whereas Adams an d
Nairne contract most of their products from other masters in the city . 1 8
Ljungberg's notebooks abound in useful intelligence . Ljungberg was o f
course well aware that his espionage and shipment of four heavy boxes o f
contraband were a violation of the tool acts . To avoid jeopardising his ow n
person he dissociated himself from his consignment and entrusted it to a
shipping agent . So, in August 1789 when it was seized in London by Thomas
Cross, a customs and excise officer, several years of hard work collecting i t
seemed to have been entirely wasted. Cross, however, was particularly interested in the notebooks since they might contain evidence proving that it s
16
Ibid .,
p . 134 .
'Ibid ., p . 133 .
18 Ibid ., p . 152 .
MfM 46 :2
57
owner had rendered himself liable to prosecution . The tool acts contained a
carrot and a stick . The carrot was a premium of fifty per cent of the value o f
the seized goods sold at a public auction . The stick was the dismissal of th e
civil servant if he let contraband slip through . Unfortunately, the notebooks
were written in gothic German, a writing and a language Cross was unabl e
to read . So he turned to Thomas Byerley for help . He was Josiah Wedgwood's partner and agent in London and his factory, Etruria, Staffordshire ,
appeared in the notebooks as one of Ljungberg's targets of espionage . Byerley reported to the Committee of Staffordshire Potters presided over b y
Wedgwood, who was alarmed and took immediate action .
Wedgwood contacted Matthew Boulton for support . Ljungberg had obtained a letter of recommendation to Wedgwood from Boulton sinc e
he knew him well as a very ingenious man and a good chymist . Ljungberg lived at Birmingham about a year and was esteemed for his in genuity, modesty and gentleman-like behaviour . But we all suspecte d
that he was employed by the Court of Denmark to collect such knowl edge in this country as might be useful in that . And although I have
some regard for Mr . Ljungberg, yet nevertheless I hope such decide d
measures will be taken as will prevent the transplantation of any o f
our manufactures . . . 1 9
Boulton and Ljungberg had in fact been on friendly terms during the latter' s
stay in Birmingham . A few months before his departure Ljungberg expresse d
his
sincere thanks for the favour and the kind reception you pleased t o
show me in Your house during my stay in Birmingham ; and also fo r
the many happy hours, which with the loss of your own time, hav e
been so very instructive to me . 2 0
To deserve Boulton's confidence Ljungberg had been an intermediary between him and Chippendale's in London, and other favours were returned a s
well . But according to Wedgwood, Ljungberg had already sneaked aroun d
at Etruria for several weeks trying to snatch information on kiln technology ,
19 Boulton's letter of recommendation of 23 .9 .1789 attached to letter of 1 .9 .1789 fro m
Neale & Byerley to the Custom's House, London ; The Treasury, T1/673, XC 14104, Publi c
Record Office, Kew.
20 Birminghaln Public Library, The Boulton & Watt Collection .
58
D . C. Christensen
MfM 46 : 2
raw materials and know-how when he handed in Boulton's letter of recommendation . Finally, however, Ljungberg had run away before he could b e
apprehended, which was Wedgwood's intention as he discovered that Ljungberg was bribing his workers to extract intelligence concerning his mode o f
production . 2 1
When Wedgwood had learned about the seizure of Ljungberg's notebook s
uncovering the wide range of targets of espionage, he planned to pass o n
this information to his fellow-victims to gain their support to persuade Parliament to have the tool acts tightened and extended . With this in mind he
had approached Boulton, and he now voiced his plan to Samuel Garbett, th e
arms manufacturer and head of the Commercial Committee of Birmingha m
who had prompted the 1785 extension of the list of contraband . 22 Wedgwoo d
aimed at nation-wide support for his claims, but to achieve this he needed a
translation of the notebooks . The intelligence collected by Ljungberg might
enable the victimised employers to prosecute their corrupt workers . It might
also reveal the names of workers whom this Danish spy had suborned or trie d
to suborn to sell their skills to a foreign power . If Wedgwood succeeded i n
uniting British manufacturers they might be able to convince Parliamen t
that Britain's industrial head start was at stake unless stricter preventiv e
measures were taken .
Wedgwood's strategy was only a partial success . He did obtain the permission to translate Ljungberg's notebooks . After all, it was in the interest
of Thomas Cross, the seizure officer, to be able to submit evidence to th e
court to have Ljungberg convicted and his premium paid out . Nevertheless ,
Wedgwood failed to unite his fellow-victims . He was very disappointed t o
learn how unaffectedly the majority of them reacted to his warnings . I have
not been able to trace the replies from manufacturers in the Wedgwood files ,
but I think it reasonable to summarise their reactions into three categorie s
on the basis of their assumed interests:
1) Wedgwood was enjoying a virtual monopoly on the world market o f
queen's ware, but his mode of production could be imitated once hi s
tools, kilns and raw materials were known .
2i-Wedgwood manuscripts 39/28404, pp . 81-88, University of Keele Library . My thank s
are due to the late Prof . John R . Harris for this reference .
22 S . Garbett's letter to W . Stiles, Commissioner of the Treasury, 25 .8 .1789, The Treasury, T1/673, XC 14104, Public Record Office, Kew .
MfM 46 :2
2)
59
Boulton ' s production of steam engines and minting presses was secon d
to none . He knew that he and Watt had a head start of thirty years o r
more, and hence it was in his interest to define a visitor like Ljungber g
as a potential customer rather than as a dangerous spy . Boulton's at titude to the tool acts changed like the direction of a weathercock . O n
the one hand he had nothing to fear from manufacturers abroad, since
his technological leadership was beyond emulation ; on the other hand
he paid lip service to the policies of those fellow-manufacturers wh o
were also buyers of his steam engines, e .g . Wedgwood and Arkwright .
3)
A large group of textile manufacturers and instrument makers appeared to be sceptical about the effectiveness of the tool acts an d
preferred to rely on their own secrecy and to take their own precautions . Examples of this are Ramsden's workshop rules and Arkwright' s
letting the dead body of an apprehended spy hang in a tree outsid e
one of his factories as a deterrent . 2 3
If this suggestion about the diverging interest of British manufacturers i s
approved, how could they possibly establish a united front vis-à-vis intruders? Hence, Wedgwood's initiative petered out . 24 Ljungberg escaped and
was never charged by British authorities . 25 Four years later he was back in
London on a residence-permit to reassume espionage .
What happened to his four boxes of tools? Well, as soon as Ljungberg wa s
back in Copenhagen he reported to his superior, Count Schimmelmann, and
Count Bernstorff, the foreign secretary, instructed the Danish-Norwegia n
consul in London to do his utmost to have Ljungberg's consignment released .
The consul was sceptical since the seizure had been reported by a newspaper
agitating the public . The consignment was confiscated, but part of it showe d
up at an auction where the consul or his puppets offered the highest bid fo r
a number of books, queen's ware and some rather innocent objects . Mos t
of the goods, however, were lost, including a lathe, some `clock-maker' s
tools', and unspecified tools and scientific instruments, estimated at £250 .
This means that Thomas Cross, the customs and excise officer, received a
premium of approx . £125, or twice his per annum salary . The seizure of
Ljungberg's boxes is one of the few known manifestations of the effect o f
23
Ljungberg's notebook (note 8), p . 223.
24 Wedgwood manuscripts 39/28404, p . 88, University of Keele Library.
25I
regret to confess that my statement about his imprisonment in Centaurus, vol. 37,
199/x, pp . 290-321 was based on conjecture and must be refuted .
60
D . C. Christensen
MfM 46 : 2
the tool acts .
What happened to the notebooks? For a long time their fate was a riddl e
to me . Had they survived at all? I searched them here ; I searched the m
there ; I searched them all and everywhere . Were they in Britain? Were the y
in Denmark? Coincidentally, I learned that a notebook by Ljungberg wa s
kept by the Public Record Office in Stockholm . But was it the notebook?
I requested The Public Record Office in Copenhagen to obtain a loan o f
it, but our Swedish friends refused the request, estimating that the ite m
was too valuable to be sent abroad . So, I went to Stockholm, repeated my
request and sat down waiting in the reading room for one hour, for tw o
hours, when an archivist informed me that this valuable source materia l
was untraceable . From my looks the archivist must have realised that I wa s
somewhat perplexed by the fashion a Nordic brother and this unique piec e
of evidence were being treated . So, the head of the Public Record Offic e
sent out his army of archivists to search several kilometres of shelves . He
called me at my hotel in the evening . Ljungberg's notebooks had surfaced .
The next morning I found myself in the reading room staring at 273 page s
mostly in gothic German and dozens of outstanding technical drawings an d
descriptions written by a master spy and last glared at more than tw o
hundred years ago by Wedgwood, the vindictive potter from Staffordshire .
Ironically, on the last page of his notebook Ljungberg had glued his lega l
residence-permit dated 1793 .
The case of Ljungberg, who has left us with probably more source material
than any other spy, shows how dangerous and troublesome it was to transfe r
technology. But even the secrecy of instrument makers and the deterrent o f
the tool acts could not prevent the know-how from falling into foreign hands .
The tube drawing machine was inaccessible to the public in contrast t o
Ramsden's dividing engine, which was at the disposal of British instrument makers only. This is why Jesper Bidstrup, of whom we shall soon
hear a lot more, was thrown upon his chamber master in London when he
wanted to divide his sextant in the most accurate fashion possible . Bidstrup
could not go to Troughton's workshop himself to have his sextant divided ,
since Troughton was only obliged to divide for British instrument makers .
Secondly, although Ramsden's dividing engine was in print, the published
drawings were inadequate as models for an instrument maker wishing t o
copy it . We know that Henrik Gerner and Thomas Bugge, both member s
of The Society for the Encouragement of Science, Agriculture, &Commerce,
MfM 46 :2
61
London, the English parent organization of The Royal Danish Society fo r
Agriculture, were in possession of Ramsden's book, but Bidstrup, operatin g
in London in the decade 1787-1798 could not retrieve a copy . 26 And even i f
he had been able to locate it he would not have been able to copy a dividin g
engine . A workshop equipped with one of Ramsden's other notorious tools ,
the screw-cutting lathe, was the necessary prerequisite . This leads us on t o
The fourth strategy consisting in sending a person to London to b e
trained in the making of instruments . In this instance Jesper Bidstrup is
the main character who has left clues of his activities in a number of pre served letters . 27 Thomas Bugge, secretary of the Academy of Science, an d
Chief surveyor, then selected him as being one of his talented students o f
mathematics and astronomy to go to London to acquire the skills of a n
instrument maker . Bugge contacted his colleague, Sir Joseph Banks, asking
him to recommend Bidstrup for an apprenticeship in one of the excellen t
workshops . This proved to be more difficult than anticipated . Banks sen t
notes to Bugge and Bidstrup that his inquiries had been unsuccessful .
London instrument makers maintained an attitude of reserve and secrec y
against teaching a foreigner a craft that had procured so much wealth and
reputation from abroad . An apprenticeship of seven years was costly eve n
for a natural-born British subject who had to pay £50-200, a sum that th e
young man would lose to his master in case he left prematurely . The Swedish
king had paid Ramsden £200 to apprentice Appelqvist, but they did no t
get on together . And although Cassini, the French astronomer, had bough t
scientific instruments from Ramsden for more than a thousand pounds ,
Ramsden refused to let any French people into his workshop .
So, Bidstrup turned towards Count Brühl, the envoy from Saxony, a wel l
known amateur astronomer and consultant, who opened the doors to tw o
of the many subcontractors in the instrument business of London, chambe r
master White, subcontractor to Nairne & Blunt for £100, and to Harbin, a n
optician and chamber master working for Dollond . After two years Bidstrup
left White in friendship to join Higgins at Walworth . Higgins was Ramsden' s
first apprentice and worked long for him till they entered into some sor t
26
Description of an Engine for Dividing Mathematical Instruments by Mr . J. Ramsden ,
Mathematical Instrument maker, published by order of the Commissioners of Longitude ,
London, 1777 .
27 The account of Jesper
Bidstrup is an abbreviated version of my article in Centaurus ,
vol . 37, 199), pp. 290-321 which contains all references and a few illustrations .
62
D.C. Christensen
Mf1VI 46 : 2
of companionship ; Higgins moved into Ramsden's house in Piccadilly an d
took over as a shop steward, but according to Bidstrup Ramsden's mediocre
moral character only allowed Higgins to stay for eight months . Higgins wa s
said to be the best supplier to Nairne & Blunt, but also worked for Adam s
and was presently completing a polar sector and the biggest transit instrument yet made in England, initiated by Sisson .
Bidstrup obtained reduced working time in Higgins' workshop by two hour s
in order to open a workshop of his own at Leicester Square (in Newton' s
old house) . Bidstrup talks to Herschel about his new discoveries and instruments and reports their conversation to Bugge . He then ships his sextant ,
an instrument of which he is very proud, divided by Troughton's engine ,
demonstrated to Joseph Banks and in Copenhagen assessed as matchin g
the best of English instruments . At Leicester Sq . Bidstrup also makes microscopes, a 4" achromatic telescope, and à mechanical power' for Hauch .
Bidstrup soon realised that in order to set up an up-to-date workshop i n
Copenhagen at least five British inventions were prerequisite, viz . the dividing engine (to reduce costs), a glass polishing machine, a tube drawin g
machine, a cupola furnace and a screw cutting lathe .
It proved most complicated for Bidstrup to have steel foundries in Susse x
and Sheffield cast the mandrels for a tube drawing machine and the variou s
discs for glass polishing . He had to make the moulds himself, which he did ,
but his journey on foot to Sussex to collect the items (he communicate d
with Sheffield via an agent) was postponed by lack of transfer of mone y
from Denmark, so that, when finally he showed up capable of paying, h e
learned that the steel mandrels had indeed been made, but since they ha d
not been collected the steel had subsequently been recast into other items ,
his moulds had been ruined and had to be reconstructed by himself fro m
scratch .
When the mandrels, now having been cast twice, were to be shipped t o
Copenhagen from Yarmouth the utmost care had to be observed, since thei r
exportation was illegal according to the tool acts, so the consignment wa s
handled by a trustworthy agent, Mr . Wheeler . Bidstrup, of course, woul d
learn from the bad luck Ljungberg had experienced in 1789 . An additiona l
complication, however, occurred . The Duke of Portland had established a
private corps of guards to enforce the tool acts, and Bidstrup, although i n
possession of a valid passport and having cleverly dissociated himself fro m
his consignment, felt obliged to hide and change his itinerary . Neverthe-
MfM 46 :2
63
less, he did escape and his tools, so arduously acquired, arrived safely i n
Copenhagen .
However, this is not a happy ending story . Bidstrup established his worksho p
at least partially, and the magnificent tools were ready to be operated .
Then, in 1802, he fell ill and died . His widow had to sell his equipment b y
auction because she owed its value to the government . Five years later th e
British bombardment of Copenhagen hit the Round Tower, and Bugge' s
library and collection of instruments were partly blown up . McKenzie, a
Scottish surveyor working on Iceland and a friend of Bugge ' s, assured hi m
that he and his fellow scientists in Britain had talked about the inciden t
with much concern and all felt convinced that the bombardment must b e
some misunderstanding . After all Bugge was a fellow of the Royal Society .
In conclusion I would like to stress two points . Firstly, instrument makin g
was revolutionised during the second half of the eighteenth century. Befor e
that time all parts of the instruments were made by hand, but from no w
on most parts were made by improved tools and machines . This transitio n
changed the division of labour in the workshop and the old guild system onl y
survived by name . Hence, transferring modern instrument making involve d
the acquisition of the innovations of the trade such as the tube-drawin g
machine, the glass polishing machine, the cupola furnace, the screw-cuttin g
lathe, and the dividing engine . It also involved the training of skills to operate these innovations, and since there was nobody capable of this in Copenhagen, where, furthermore, the old guild system still prevailed, these skill s
had to be appropriated by one person . Secondly, the cases of Ljungberg
and Bidstrup indicate that no matter how secretive manufacturers wer e
and no matter how carefully the tool acts were observed by determine d
civil servants they did not succeed in preventing their precious technolog y
from falling into foreign hands . All they could do was to make the life o f
predacious spies and trainees more troublesome and dangerous .
MfM 46 :2
65
Instrument Maker on the Run :
A Case of Technology Transfe r
Olov Amelin *
Abstrac t
This paper will deal with a number of design principles and a n
instrument maker . The most important parts are played by a circle ,
the number 96 and the craftsman Johan Ahl (1729-1795), who made
the instruments that were used by the Norwegian surveyor and math ematician Caspar Wessel.
Whether Ahl is to be regarded as a deserter or not depends on th e
view one takes of the relationship between the Royal Swedish Academ y
of Sciences and the Royal Danish Academy of Sciences and Letters
in the 1760's . The question of technology transfer can also be see n
from two points of view . From a Swedish point of view it was highly
undesirable ; from a Danish point of view it was most welcome and
wanted .
Introductio n
Johan Ahl came to the workshop of the instrument maker Daniel Ekström i n
Stockholm in its heyday around 1750 . Most likely he was one of the "mathematically trained workmen" who together with three younger apprentice s
were in the workshop in 1754 . 1
After the death of Ekström in 1755 there was no obvious heir to the business . Various solutions were discussed, but it appears that Ekström himsel f
* The Nobel Museum, P .O . Box 2245, SE-10316 Stockholm, Swede n
' The tax records of 1754, City archive in Stockholm .
66
O . Amelin
MfM 46 : 2
wanted Johan Ahl and Johan Zacharias Steinholtz, one of the other appren tices, to take over the workshop, which was actually owned by the Roya l
Swedish Academy of Sciences . The Academy agreed with Ekström and appointed Steinholtz as director and Ahl as his partner .
However, joint management by Ahl and Steinholtz does not appear to hav e
worked successfully. Steinholtz took over the business and Ahl found him self making more simple instruments such as thermometers, compasses and
drawing instruments . 2 In 1760 we find Ahl mentioned in the tax register
of "Lower West Klara" , an area in central Stockholm . 3 He is mentioned a s
"The partner in the workshop Johan Ahl, married, 1 powder, 30-31 year s
of age" . The living quarters and the workshop were located in the bloc k
Kajan No . 49, which also included a herb garden . Ahl and Steinholtz had
moved from Ekström's old workshop in the observatory to new premise s
in Stockholm . Altogether twelve people worked here, including Ahl's partner, the three-year-younger Steinholtz, and Steinholtz' wife . Two maids, five
apprentices and a lodger are mentioned in the register . The information " 1
powder" indicates that they could afford the luxury of powdering their wigs ,
for which a tax was paid . But the wealth was illusory . In March 1760 the
relations between Ahl and Steinholtz had deteriorated to such an exten t
that the Academy thought it would be best to separate them . Financially,
Steinholtz benefited from the separation . Two-thirds of the allowance, which
since Ekström's day had gone to the workshop, now stayed with Steinholtz .
Conditions for Ahl were difficult, as may be seen from the Academy minutes .
By May 1760 he was having to ask for 600 copper dalers in advance, which
he was granted . 4 Ahl's requests to be paid in advance appear frequently i n
the minutes and suggest that his terms of employment were poor . 5 However Ahl probably owed money to others besides the Academy and he wa s
also dabbling in high finance and was defrauded by a clerk at the Bank o f
Stockholm . 6
2 Lindroth, Sten, Kungl Vetenskapsakademiens historia 1739-1818, p . 799, Stockholm
1967.
3 Fataburen 1915, Stockholm 1915 .
4 The Royal Swedish Academy of Sciences, Minutes 21 .5 .1760 .
'Ibid . 21 .5 .1760 (600 copper dalers), 14/10 1761 (2000 copper dalers), 18 .11 .1761 (150 0
copper dalers) .
5 The Royal Swedish Academy of Sciences, Secretary ; Paper 33 :1 Johan Ahl, Memorandum, not dated but probably 1763 ; Promissory note signed by Colliander, Stockhol m
8/10 1760 . Colliander returns to the bank and Ahl tries to pay his debts to the Academy
MfM 46 :2
Instrument Maker on the Run
67
The last notes we can find in the Academy minutes about Ahl's work i n
Stockholm date from 17 March 1762 . The Academy's experimental physicis t
Johan Carl Wilcke demonstrated an inclination compass, an instrument fo r
vertical measurement of the earth's magnetic field, that he had designe d
and Ahl had manufactured .
The next note in the minutes tells of Ahl's escape from his creditors . Th e
Danish government promised the Swedish minister in Copenhagen to pa y
Ahl's debt to the Academy . This is as early as June 1763 . In other words ,
Ahl must have established himself quickly in Denmark and managed t o
make contact with the right people . Little is known about his departure ,
but in June 1762 Ahl is mentioned in a memorandum written by Henri k
Hielmstierne, secretary to the Royal Danish Academy of Sciences and Let ters in Copenhagen . According to the memorandum he had "[ . . . ] wit h
great difficulty arranged for immigration of a very skilful instrument make r
from Stockholm" 7 (author's translation) . The letter does not tell what thi s
difficulty was . Maybe Ahl had hesitated, but more important was the fac t
that Ahl's escape to Copenhagen meant that Sweden lost one of her tw o
most competent instrument makers, also one in whom the Royal Swedis h
Academy of Sciences had invested large amounts of money . The state o f
Swedish instrument-making was poor, even before Ahl's disappearance . Important knowledge had now fallen into the hands of a foreign country . The
Danes, on the other hand, had everything to gain . On 26 June 1761 the
Royal Danish Academy of Sciences and Letters had been granted funds b y
the King for the cartographic surveying of Denmark and the publishing of a
nation-wide map . Behind the project were the professor of mathematics and
member of the Academy Christen Hee (1728-1782), and the young assistan t
at the Round Tower Observatory Thomas Bugge (1740-1815) .
The need for surveying instruments was great in Denmark after the Kin g
had approved the project . At this time Muth was the royal instrumen t
maker, but he was considered too old and too slow to be able to provid e
and Steinholtz using a promissory note . The promissory note was not accepted by th e
Academy, see 33 :1 Utdrag af kongl. seereteraren Faæels skrivelse till h :s ex :ce . . .
Cantzlie ilresidenten grefve Ekeblad dated Copenhagen 20 November, 1763 .
7 Memorandum 27 .6 .1762 from H . Hielmstierne to J . L . Holstein, archive of the Roya l
Danish Academy of Sciences and Letters, Special archive, Geografisk landmaaling 17611765 . Written at the top of the document in another hand is a statement about salary fo r
Ahl (150 Rdlr) paid as from 21 May 1 .762 . From these sources we can date his "escape"
to Denmark to April or early May 1762 .
68
0 . Amelin
MfM 46 : 2
the Royal Academy of Sciences and Letters with the necessary equipment . 8
Johan Ahl, or Johannes Ahl as he was called in Danish, came to Copenhage n
in the spring of 1762 . He immediately started to provide the Academy wit h
instruments . Ahl must have been asked to come to Copenhagen to provide
Bugge with equipment . The Swedes' talk about desertion therefore gives a
misleading impression of the course of events . It is likely to have been an
easy decision to make for Ahl, who was burdened with debt, and also o n
poor terms with his former partner Steinholtz .
Furthermore, orders placed by the Royal Danish Academy of Sciences an d
Letters had been received in Stockholm as early as the winter of 1761-62 .
Parts of them had been carried out, probably by Ahl and Steinholtz, an d
delivered to Joachim Otto Schack, the Danish agent in Stockholm . Some
of the larger instruments, though, probably made by Steinholtz, had no t
been delivered . The Royal Swedish Academy of Sciences, with its minin g
consultant Daniel Tilas and its secretary Per Wilhelm Wargentin leadin g
the way, retained the ordered instruments, to try, if possible, to force the
Danes to return Ahl to Stockholm . In Denmark Ahl immediately received an
annual salary of 100 Rigsdaler, which was raised after a couple of months t o
150 Rigsdaler . By 1769 Ahl . was receiving 300 Rigsdaler a year plus another
50 Rigsdaler for the employment of an apprentice . 9 Ahl quickly reached
the position of being the foremost instrument maker in Denmark, and was
spoken of as "[ . . . ] our skilful Ahl" or "[ . . . ] our skilful and deserving
Ahl" .1 o
Ahl's debt to the Royal Swedish Academy of Sciences was paid off in 1764 ,
when the Swedish envoy, Mr . Sprengtporten, informed the Academy membe r
Count Ekeblad, that "after long drawn out proceedings" he had received the
4000 copper dalers that the departed Ahl owed the Academy. For his work in
connection with the recovery of this debt Sprengtporten received a specime n
s Ibid: "og den her værende naunlig Muth som har aarlig pension af Hs Majesteet ,
findes deels at være meget uaccurat og langsom deels kostbar med Hs arbeide, det og oft e
forefalder at der gaaer Slid og fall paa deslige Instrumenter, som i en Hast maa giøres til
Rette for ei at sinke arbeidet . "
9 C . Molbech, Det Kongelige Danske Videnskabernes Selskabs Historie 1742-1842 .
Copenhagen, 1843, p . 77 .
' °The information on Ahl's stay in Denmark is, unless otherwise stated, taken fro
m
Keld Nielsen, Hvordan Danmarkskortet kom til at ligne Danmark, Videnskabernes Selskab s
opmåling 1762-1820, Foreningen Videnskabshistorisk Museums Venner, Århus, 1982 .
MfM 46 :2
69
of every medal minted by the Academy . '
1 The mapping of Denmar k
The implementation of this project has been described carefully by
Nielsen, but a brief reference to the methods used may be useful .
Kel d
The Danish mapping project was large and ambitious . It undoubtedly made
Denmark one of the leading nations in the area of land surveying . The aim
was to make a nation-wide triangulation net . The method used came fro m
France, where it had been in use since Jean Picard (1620-1682) had mea sured a part of the meridian through Paris and Amiens in 1669-1670 . Th e
Cassini family then used this method when the mapping of France starte d
in 1747 . It involved connecting dots along a meridian with a series of triangles with carefully measured angles, after which it was enough to carefully
measure the length of one side of one of the triangles - the baseline . Wit h
simple trigonometric calculations, the lengths of all other sides of the triangles could then be calculated and finally also the length of the meridian .
The disadvantage of this method was that any mistake in measuremen t
was transferred to the final determination of the meridian . This made extremely accurate angular measurements absolutely essential, and the instruments had to be manufactured and used with great skill . To meet thi s
need, Denmark began an extensive training programme, in which land surveyors were taught to perform the triangulation measurements . The whol e
project was described by Thomas Bugge in 1779 in Beskrivelse over den
Opmaalings Maade, som er brugt ved de Danske geographiske Karter; med
tilføiet trigonometrisk Karte over Siveland, og de der henhØrende Triangler,
beregnede Longituder og Latituder, samt astronomiske Observationer. 1 z
2 The Ekström circle in Denmar k
Johan Ahl was trained in Ekström's workshop, a workshop that can be sai d
to have specialised in land-surveying instruments . At least this was the are a
11
The Royal Swedish Academy of Sciences, Minutes 9 .1 .1765 .
description of the surveying method used for the Danish geographical maps wit h
an attached map of Zealand, including the triangulation net, calculated longitudes an d
latitudes, and astronomical observations (author's translation) .
12A
70
O. Amelin
MfM 46 : 2
in which Ekström made his most significant technical innovations with the
introduction of the levelling tube and the circumferentor, also called the
Ekström circle . The instrument was much appreciated in Sweden and wa s
Ekström's most successful invention . The circle was presented to the Roya l
Figure 1 : Daniel Ekström's geographical circle ; illustration from his article in the
Transactions of the Royal Swedish Academy of Sciences 1750 .
Swedish Academy of Sciences in 1750 and was described in an article in the
academy's transactions the same year . 13 An interesting detail was that i t
incorporated elements of a Danish tradition of instrument-making .
In England and France parts of circles (quadrants, sextants etc .) were preferred for different angular measuring instruments . Ole Römer (1644-1710 )
on the other hand favoured complete circles and designed his own instruments of this type . Complete circles were more stable and the scales wer e
' Ekström, Daniel, Beskrifning på Et nyt Geographiskt Instrument, Kungliga Veten skapsakademiens Handlingar 1750 (Transactions of the Royal Swedish Academy of Sciences), Stockholm 1750 .
MfM 46 :2
71
easier to provide if the division of the scales could be checked against th e
diagonal of the circle . Whether or not Ekström was influenced by Römer's
designs when he made his circle is not clear . In 1763 Ahl had made a circumferentor for the Royal Danish Academy of Sciences and Letters, fo r
measuring the angles of the triangles mentioned earlier . The instrument
was much appreciated by Bugge, because it was so easy to handle . In his
description of the instrument he says that the right angles were divide d
both into 90° and into 96 sections, in other words using George Graham' s
method of dividing the circle by bisection . 1 4
The scales on the smaller circular instruments (to which category the surviv ing examples belong) were probably divided by using a template and bea r
no traces of division into 96 sections . The instrument depicted in Bugge' s
book differed on several accounts from the original design by Ekström . The
stand was more stable and the circle was provided with a semicircle with a
cog positioned at a right angle to the surface of the large circle . This made it
possible to place the circle at any angle between 0° and 90° by using an adjustable screw . The scales could be read, according to Bugge, by using fou r
vernier scales . At both ends of the adjustable tube there was a vernier scale
for the 90° scale and one for the scale divided into 96 sections . With these ,
readings as small as 15 seconds of arc could be taken . As a complement
14 Thomas Bugge, Beskrivelse over den Opmaalings maade som er brugt ved Dansk e
geographiske karter, Copenhagen, 1779 .
The circle made by Ekström, now in the Old Observatory in Stockholm, does not hav e
a scale divided into 96 sections . In Denmark three circumferentors are preserved . Th e
one mentioned in the text, signed by Ahl and later modified by his apprentice Jespe r
Bidstrup who added more bubble levels, does not have a scale divided into 96 section s
(Bugge is referring to Bidstrup's work with Ahl in a letter to J . Banks in 1787, see N y
Kgl . S . 287.1, 4°, Royal Library, Copenhagen) . The instrument is now in the Ole Römer
museum in Taastrup . Another instrument, much like the one mentioned earlier, is at Kort
och matrikelverket in Copenhagen . There is no sign of a scale divided into 96 sections
on this instrument either . The third instrument at the Aalborg University Centre is o f
a somewhat simpler and probably earlier construction . It is reminiscent of the original
instrument made by Ekström . Even the construction of the stand is identical with that o f
Ekström . All three of these instruments were probably made by Ahl . Yet another circle by
Ahl is at the National Administration of Shipping and Navigation in Norrköping . Ther e
is no trace of a scale divided into 96 sections on this instrument either .
On both contemporary illustrations of Ahl's circular instruments in Thomas Bugge' s
Beskrivelse over den Opmaalings Maade . . . (table 1) and Observationes Astronomica e
Annis 1781, 1782 f 1783 (table VIII) (in the latter a somewhat larger circle mainl y
intended for astronomical use) the scale is shown as divided into 96 sections .
72
O . Amelin
MfM 46 : 2
to the vernier scales there was also a micrometer placed in the adjustabl e
tube . Ahl made his circle in different versions, a simple variant much lik e
the one Ekström had constructed and a couple of more advanced version s
both for geographical and astronomical use . The large circle used by Bugge
has disappeared ; however, if we compare Bugge's description with the tw o
surviving geographical circles in Stockholm and Uppsala signed by Johan
Zacharias Steinholtz, there are several similarities . The adjustable tube has
a micrometer and the adjustment of the circle is the same as on Bugge' s
instrument . The stand is also similar to Bugge's . Steinholtz' instrument No .
11 was made in 1761 when Ahl was still in Sweden . Today the instrument
is at the Royal Swedish Academy of Sciences in Stockholm . Instrument No .
10 is in the Observatory Library in Uppsala . Most likely Ahl participated in
the making of No . 10, even though Steinholtz signed it . The improvement s
to the Ekström circle described by Bugge had therefore already been developed when Ahl decamped to Denmark . Perhaps some of the instrument s
preserved in Sweden were even originally made for Denmark as parts of th e
order executed but not delivered by the instrument makers in 1761 .
Bugge touched again upon Ahl's circle in an account of his visit to Paris i n
1798-1799 as a member of the "international" commission of weights an d
measurements . There he had come into contact with the repeating circl e
made by Borda . Bugge wrote that the instrument in question which had a
two-foot radius, gave very good results, but that the reports of its accurac y
were exaggerated . After analysing a few of the results he found a measurin g
error of six seconds of arc . Bugge writes proudly that circle instruments o f
similar design have been used in Denmark for nearly 40 years . He describes
some design differences, but goes on to say that in his opinion Ahl's circula r
instrument generally gives the same accuracy of reading as those of Borda . 1 5
Bugge's description of Borda's instrument is both interesting and revealing . 16 There were major differences in design between Ahl's and Borda' s
scales . Ahl's best instruments had two scales, one 90° scale and one divide d
into 96 sections . The technique was that employed in the making of quad rants, using a pair of compasses to divide smaller and smaller angles . This
was the method developed by Graham in the 1720's, which Ahl had learne d
from Ekström in Stockholm . In Borda's case there was a scale on which th e
quarter circle was divided into 100 sections . Bugge saw this as something
15 Thomas Bugge, Reise til Paris i Aarene
1798 og 1799, (Copenhagen, 1800, pp . 553) .
16 Allan Chapman, Dividing The Circle
(London, 1990, p . 118) .
MfM 46 :2
Figure 2 : Ahl's geographical circle de-
73
Figure 3 : J .Z . Steinholtz's geographi-
picted in Bugge's "Beskrivelse over
cal circle made around 1761 . Now i n
den Opmaalings 114aade . . . ", 1779 . the collections of the Royal Swedis h
Academy of Sciences . (Courtesy of the
Royal Swedish Academy of Sciences) .
74
O. Amelin
MflVI 46 : 2
negative, but Borda most probably divided his scale with more moder n
tools, i .e . some kind of dividing engine giving high precision without an y
need for another scale . When Bugge visited Paris in 1798-1799 he foun d
circles that were divided by machines equipped with microscopes . These
were also read with microscopes equipped with micrometers, giving a muc h
higher precision than Ahl's instrument . Both Bugge and Ahl had missed
out on this development . Circles and, more particularly, the technique fo r
dividing the scale continued to develop elsewhere in Europe .
3 Ahl's Quadrant
The Danish triangulation network that was measured with the circumferentor was orientated with the meridian passing through the observatory a t
the Round Tower in Copenhagen . The position of the meridian was fixed by
solar observations . To orientate the meridian in the field Bugge measure d
the angle between the meridian in the northern direction and the line o f
sight towards the south spire of the cathedral in Roskilde . Another line, a t
right angles to the meridian through the observatory, "the Perpendicular" ,
was measured to enable the construction of what we would call a co-ordinat e
system with an x-axis and an y-axis . We shall not go deeper into the techni cal details of mapping, but the two lines through the Round Tower and th e
triangulation net made up the framework for the whole mapping project .
To be able to give the map the exact degree of longitude and latitude i t
was necessary to perform a very careful determination of the Round Tower .
For this purpose Bugge ordered several instruments from Ahl's workshop ,
one of which was a 3-ft quadrant (92 .5 cm radius) delivered in 1778 . With
this quadrant Bugge could determine the polar altitude of the observator y
after measuring 28 fixed stars during the winter of 1778/79 . Complementary measurements were also made using a circumferentor . The result that
Bugge arrived at was 55° 40'56", which may be compared with the moder n
measurement of 55° 40' 52 " 17 . The work was done mainly with the quadran t
made by Ahl . Once again we can find similarities with instruments that Ah l
had seen or worked with in Stockholm . The stand was similar to the on e
made for the quadrant delivered to the Stockholm Observatory in 1757 . Th e
17 Nielsen, p . 40
. Nielsen states that Bugge had calculated the value 55° 40' 52" bu t
changed the seconds to 56", because it gave a more even number when working wit h
six-digit logarithms . Log sin 55° 40' 56"=-0,083060, an even and "handier" number .
MfM 46 : 2
Figure 4 : Quadrant made by Ahl in
1778 .
75
Figure 5 : Quadrant made by Bird in
1757 . (Courtesy of the Royal Swedish
Academy of Sciences) .
maker of this instrument was John Bird, the most famous of all instrumen t
makers at the time . The size of the instrument differs by six inches fro m
Bird's construction . One important point of difference is in the metal frame work ; Ahl's design has squares, whereas Bird's has spikes radiating from a
central hub around which the quadrant could be turned . Both instrument s
had two scales, one divided into 96 sections and one 90° scale . Ahl's quadrant was provided with an adjustable tube with both a vernier scale and a
micrometer . Today the instrument is at the Round Tower . Tubes and other
O . Amélin
76
MfM 46 : 2
parts are missing .
The longitude was of course harder to determine exactly, but based on th e
eclipse of the moons of Jupiter the position of the Round Tower could b e
determined in relation to the Paris Observatory and the Stockholm Observatory.
4 The restoration of the Observatory at the
Round Tower - a comparison with the Stock holm Observatory
At the time of these measurements Bugge was busy restoring his observator y
at the Round Tower . A new top storey with an octagonal room in the middl e
and two wings was built on the roof of the tower . Ahl continued to receive
orders for more instruments . In addition to the quadrant, Ahl was asked t o
make a zenith sector, a mural quadrant and a transit instrument . A Swedis h
traveller, the astronomer Henrik Nicander, visited Bugge in the late summe r
of 1778 and reported back to Wargentin about the work on the observatory :
The largest observatory room is a hexagon [finally the room was buil t
like an octagon, author's note], with a window to the north and south .
Inside a mural quadrant with an 8-ft radius will be placed . It has not
yet been built, but has been ordered from the skilled instrument make r
Ahl, who is residing in Copenhagen . A sector with a 16-ft radius will
also be installed, but in such a way that it is placed in a hole from th e
bottom of the tower, all the way through the hexagon . Through this
hole the stars can be observed clearly during daytime . On the sides of
the hexagon, to the east and to the west, are two rooms . In the roo m
to the east, a 6-ft transit instrument by Ahl will be placed on a marbl e
pillar . On another pillar in the same room the observation clock ha s
been placed . In the other room to the west, a quadrant with a 3-ft
radius will be placed, which is now in the hands of Ahl [ . . . ] The
view from all sides extends for miles and to adjust the mountings o f
the instruments on the meridian a sign will be placed on the beach o f
the island of Amager - a Swedish mile [10 km] from the observatory . 1 8
18 Nicander to Wargentin 5 .9 .1778, The Royal Swedish Academy of Sciences .
MfM 46 :2
77
Bugge built the best-equipped observatory in northern Europe . The prototypes were mainly the English observatories that he had visited in 1777 .
The ambitious nature of the project forced the instrument maker Ahl t o
work on very demanding tasks, and he was able to develop his skills . I n
the letter quoted above, Nicander sends Ahl's compliments and states tha t
Ahl "seems to be more appreciated in this place, than he was by us" . Almost a year later Bugge himself writes to Wargentin and tells of his plans .
Either Nicander had misunderstood some matters or Bugge had adjuste d
sizes and positions of the instruments . The mural quadrant was suppose d
to have a 6-ft radius and the zenith sector a 12-ft radius, not 8- and 16-ft
respectively as stated by Nicander . Bugge does not mention anything abou t
placing the zenith sector in the centre of the tower . The zenith sector was
placed together with the other instruments in the observatory building o n
the roof .
Bugge's observatory contained all instruments that were used at this tim e
in the large European observatories . A comparison with Wargentin's set o f
instruments in Stockholm shows that Bugge made greater use of precisio n
angle instruments . During his tour of England he visited several of the mos t
famous instrumentmakers and watchmakers in London . Bugge was able t o
learn the design details both at Edward Nairne's and at Jesse Ramsden' s
workshops . At the observatories in Oxford and Greenwich he was given th e
opportunity to make observations using both transit instruments and mu ral quadrants . 19 In the diary that Bugge kept during his tour, there ar e
details of the instruments with drawings and comments . This knowledge
was passed on to Ahl upon Bugge's return and we can find similarities between the instruments constructed by Ahl, the drawings in Observatione s
Astronomicae Annis 1781, 1782 and 1783, and the English instruments de picted in Bugge's sketches . 20 Ekström had also made similar instruments ,
and Bugge's contribution can be seen as a refinement of details . Ahl kne w
the basic principles already .
In Bugge's letters to Wargentin he describes several technical details of th e
instruments . The sight tube in the movable quadrant and the transit instrument had an achromatic lens system . The scales were divided into 96
19 K . M . Pedersen, "Uddrag af Thomas Bugges Dagbog 1777", Festskrift i Anledning af
Universitetsbibliotekets 500 Års Jubilaurn (Copenhagen, 1982) .
20 Ibid . pp . 155 . and Bugge, T . Observationes Astronomicae annis 1781, 1782 é1 1783,
Hauniae 1784 .
78
O . Amelin
MfM 46 : 2
sections and 90° . A special construction to turn the movable tube on the
quadrant was designed by Bugge himself . He took a great interest in the instruments and participated in the development and construction work wit h
Ahl in a way that Wargentin never did with Ekström . This also depended
on the difference in scope between Bugge's and Wargentin's work . Fro m
the beginning Bugge worked with observations where precise calculations o f
position were of great importance . That Bugge had great confidence in hi s
instrument maker is well expressed in the following letter :
All these instruments have been made by Mr Johannes Ahl, Swedis h
by birth, whom I respect highly for his competence . I do not thin k
that he will yield to any instrument-maker in Europe except to th e
deceased Bird and the yet living Ramsden . 21 (author's translation) .
Wargentin's studies of the moons of Jupiter were not dependent upon an gular measurements in the same way . But there was also a basic differenc e
in attitude between the two astronomers ; this can clearly be seen in Wargentin's reply, written in November 1779 :
I have not wanted to recommend the Royal Swedish Academy of Sciences to purchase such expensive instruments, because it is hardl y
worth the effort in our climate, where the opportunities to observe ar e
very few . The three to four months of summer, are one long day, whil e
the winter is either cloudy or so cold that one cannot handle such sensitive instruments with a steady hand . Also I think it is unnecessar y
to put so much money into astronomy when so many observatories i n
Europe, in better climates, are already equipped with the finest instruments and the best observers, which I could never hope to equal, muc h
less surpass . They all have the same sky, the same work . When one
does not think one can do better than them, less joy and honour come s
out of the cost, the watching, the effort . 22 (author's translation) .
The halcyon days of the Round Tower end around 1800 . Thereafter Bugg e
had to devote more time to military work in connection with the war wit h
England, and his astronomical work was therefore neglected . The instruments fell into decay and most were removed from the Round Tower . I n
1808 The Copenhagen Fire commission decided to convert the tower int o
21 Bugge to Wargentin
22 Wargentin to Bugge
20 .6 .1779,
2 .11 .1779,
The Royal Danish Academy of Sciences and Letters .
The Royal Library, Copenhagen .
MfM 46 :2
79
a fire tower . 23 But Bugge stopped the plans and his building was able t o
continue to work as an observatory until the end of the 1850's .
5
Summary
When Ahl left Sweden, Swedish instrument-making was on the brink of ruin .
In Denmark on the other hand, it began to flourish . Why ?
Denmark had all that was required for the successful co-operation betwee n
instrument makers and scientists that was essential . In Sweden this cooperation lapsed when Ekström died . Wargentin did not have confidence i n
Steinholtz and Ahl, so large orders were placed abroad .
Was Ahl a deserter ?
From a Swedish point of view Ahl's move to Denmark was most unfortunate .
Every attempt was made to get him back . His departure caused both ex pense and the loss of valuable knowledge to Sweden . From a Swedish point
of view the transfer of knowledge was therefore highly undesirable . Denmark, on the other hand, no longer had to import surveying instruments .
Instead it became one of the largest manufacturers . The investment in Ah l
enabled Denmark to build the most advanced instrument-making worksho p
in Scandinavia . The large mapping projects, both the theoretical and th e
technical aspects, placed Denmark far ahead in the field of cartography .
Ahl's work is an interesting and clear example of how technology coul d
be transferred . First from English instrument makers to Ekström, then t o
Ahl, who took his knowledge with him to Denmark, where Bugge adde d
the knowledge he acquired on his tour to England . The circumferentor, or
Ekström circle, was a unique construction heralding theodolites and repeat ing circles . But the decisive factor in the successful cartographic project wa s
the co-operation between artisans and scientists . Without this, the scientifi c
image of Denmark would not have received the boost that it did .
23 Kjeld Gyldenkerne, Per Barner Darnell and Claus Thykier,
Dansk ast7onomi gennem
firehundrede år (Copenhagen, 1990), p . 103 .
MfM 46 :2
81
Wessel as a Cartographe r
L . Kahl Kristensen *
Abstrac t
The primitive level of geodesy at the time the young Caspar Wesse l
became assistant surveyor is described . Fundamental geometric concepts were not yet developed and the shape of the Earth still debated .
Many computations tacitly assumed a flat Earth . When Caspar Wessel took over the triangulation of Jutland he proposed a division of the
country. His plan was approved by the Royal Danish Academy of Sci ences and Letters at a meeting on February 5 1779 . Although this plan
is often mentioned, the content of it is not known . Inspection of
the published maps, however, shows that it was a rational division, stil l
in use, which allowed the sheets to be joined . This system is continue d
to the present day.
The kind of applied mathematics which filled Wessel's professional lif e
is indicated by Thomas Bugge's writings .
A new geometric theorem shows simply, in terms of the Gauss cur vature, the size of the error in regarding the Earth as flat . This "gap
theorem" is invented for the present paper .
1 Introductio n
Soon after the mathematically talented Caspar Wessel arrived in Copenhagen to study law he was employed (from 1764 to 1805) by the Roya l
Danish Academy of Sciences and Letters in the surveying of Denmark .
This project was an imitation of the mapping of France under the auspices of
the French Academy . It involved practical work in the countryside, applie d
` Institute of Physics and Astronomy, University of Aarhus, DK-8000 Århus C, Denmark
82
L.K. Kristensen
MfM 46 : 2
mathematics and much computing . The kind and level of this mathematics ,
on which Wessel must have spent much time, is shown in Chapter 2 . Th e
leader of the mapping was Thomas Bugge (1740-1815) whose publishe d
works show the state of the art of geodesy at this time .
In Chapter 3 the cartographical work of Wessel is shortly discussed . H e
continued Bugge's triangulation but used a much more rational system fo r
the division of the maps .
Finally, in Chapter 4, we mention the improvements after the epochs o f
Bugge and Wessel, when the topographical survey became the responsibilit y
of the General Staff of the army . Again the project was copying an earlier
French project, but Wessel's rational division of the maps was continue d
and in some respects the maps even surpassed the French archetype .
2 The epoch of Thomas Bugge
In 1762, the Royal Danish Academy started an 80-year work on the mappin g
of Denmark in scale 1 :120000 by engaging two surveyors, one of whom wa s
Bugge . At the age of 19 he had assisted in a plane table survey where th e
position of the table was determined by a triangulation on the table itself .
This gave the dearly bought experience that error accumulation rapidl y
distorted areas at a distance from the base lines . After this Bugge became
a devotee of what he called "the method of parallel lines" . Here paralle l
lines were marked out by sticks in the field at a distance of 10000 alen' an d
measured by a chain . The intention was that these lines should serve as a
frame and be stations for the tables . In Zealand the parallel lines were a s
long as 200000 alen and aligned by sticks at a distance of 30-70 alen, an d
by higher signals at distances of the order of 1000 alen . The longest line s
thus required the placing of a stick about 4000 times !
Bugge discussed observational errors very carefully. He distinguished between random and systematic errors, and estimated instrumental errors by
the widths of threads and division lines . He also made psychophysical experiments [4, p .16] with different people to determine the visibility of signal s
under different illuminations and determined the resolution power of the
' Measures used : 1 alen = 2 fod (Rhine or Leyden feet) = 0 .6277 m . 1 favn ("fathom" )
= 3 alen = 1 .8831 m . 1 mil = 12000 alen = 7 .532 km .
MfM 46 :2
Wessel as a Cartographer
83
human eye to +1' in clear sunshine (from behind) and to ±2' under less
favourable conditions .
When possible, Bugge, after critical discussions, combined several indepen dent determinations of the basic parameters . For the latitude of the Royal
Observatory in Copenhagen (the "Round Tower") he even included Jea n
Picard's approx. 100 years older data . If the critical examination did no t
indicate that some data should be rejected, he used simple averages . He
warns explicitly against the use of "raisonnerte" averages or a preferenc e
for values which happen to appear twice . Very reasonably he argues that ,
by chance, such values could be most in error [1, p .85] .
Due to his expertise, Bugge was ordered by the king [5, p .12] to be a member of the International Commission on the Metric System . He pointed out
that the definition of the metre in terms of the meridian quadrant could, a t
best, be realized to +0 .1mm . Today we know he was right : the prototyp e
metre is actually 0 .2 mm too short! On a commission with starry-eyed theorists Bugge represented common sense . Among the radical proposals of th e
Commission were, for instance, that a day should have 10 rather than 1 2
hours, a mariner's compass point should be 1/10 rather than 1/8 of a right
angle, and even the currency should be reformed to the metric system ( 1
franc = 5 g 90% silver) . With his great practical experience Bugge pointe d
out that such reforms would cause much confusion, be very expensive an d
take a hundred years to be accepted and in common use . Fortunately, in hi s
opinion, there was no need to introduce this system in Denmark . From the
Elbe to Nordkap we had, since 1682, a uniform system of weights and mea sures [5, p .539] which was constructed by Ole Romer, one of his predecessor s
as Astronomer Royal .
Despite his sound attitude to a theory of errors, Bugge badly lacked ou r
days' statistical theory for the accumulation of errors . From 244 repeate d
weighings [5, p .632] he found, as mean value, 1 kg = 2 .002769 Danish pound s
and he estimated the error to be less than +0 .000002 pound, but the concepts of "probable" or "mean errors"- could not be used as they wer e
not yet known to Bugge .
The want of a statistical theory of errors is clearly illustrated in his discussion of the accuracy of staking out straight lines [4, p .20] . The sticks shoul d
be sighted equally from both sides "so that the small errors would be no w
on one side, now on the other, and hence, ultimately the line will approach
mathematical accuracy" . If trees were in the way of the line they should be
L.K. Kristensen
84
MfM 46 : 2
passed by parallel lines alternatively to the right and the left of th e
obstacle . These displacements were allowed because the diameter of a tre e
is too small to be drawn in the scale of a map !
Assume a sighting error ±2' and a 1000 alen distance between the signals .
Bugge is aware that each stick introduces an error so their number shoul d
be kept small by making the distances between them as large as possible .
He could not know that the transverse error would increase as the inverse
square-root of this distance and, for a main line across Zealand of 20000 0
alen, ultimately become :
2x
3438'
x 200000 x
3 ô=
OOOÔ
1900
alen
(1 )
corresponding to ±10 mm on the final map . Bugge was aware of an intolerable error but was so confident in his alignment method that he ascribed it
to the 1/2-1% shrinkage of the paper when the draft maps in scale 1 :2000 0
are cut from the plane tables . With his usual care, [1, p .5] this shrinkage
was investigated as a function of time and humidity.
We cannot tell if it was expected that the "method of parallel lines" could
stand alone and only needed a triangulation as a supplementary check . The
original plan from 1761 [1, p .VIIl] included a supporting triangulation, bu t
the actual work started after the plane table survey. The sides of the triangles should not be less than about 2-3 mil and their angles were measure d
with a full circle with radius 1 fod constructed by Johannes Ahl (17291795) 2 . In a critical discussion of this instrument [1, p .21] Bugge estimate d
its accuracy to 8-10 " , in good agreement with a mean error ±9" estimated
from the sum of the angles in his triangles .
Bugge's 80 main triangles on Zealand mainly followed the coast line and
avoided the central part of the island . Angles less than 30° were measure d
several times if they could not be avoided . The angles were adjusted by
judgment so that their sum in each triangle became 180° exactly. If th e
angles were considered equally well determined they obtained the same correction . 3 Haze or bad sightings during observations resulted in larger correc tions . Small angles obtained smaller corrections, maybe due to the greate r
2 See Olov Amelin's paper in this volume .
3 These corrections are due to observational errors . Legendre's theorem, published in
1772, was not used by Bugge . The spherical excess correction in Bugge's largest triangle ,
No . 79 [1, p .75}, is only 0,7 " . Incidentally, no correction at all was made in this triangl e
No . 79 . This indicates that one of the angles was computed .
MfM 46 :2
85
care in their measurement . A significant number of triangles (14 of 80) ar e
not adjusted, indicating that one angle may have been computed . Not al l
geometric constraints were satisfied by this procedure . 4 It appears, however ,
to be a very reasonable method prior to the invention of the method of leas t
squares .
Without any estimates of the degree of approximation involved, Bugge con sidered the Earth flat in his computation of the triangles . A justification he
could have had in mind is the small difference between arcs and chords . In
his textbook on surveying [4, p .21 he compares an arc s = 7 .505 geographical
miles on a sphere with radius R = 860 miles with the corresponding chord .
Curiously enough he finds the difference arc minus chord to be 0 .00012 4
miles, or five times the correct result 0 .000024 (= s 3 /24R 2 ) . This error i s
due to excessive roundings and appears unchanged in the 1795 and 181 4
version of the text .
Let us now prove that the flat Earth approximation can be justified fo r
Bugge's task of producing a topographic map . This will be done by th e
useful, though rarely used, "gap theorem" probably formulated and prove d
here for the first time :
From a point O on a spheroid a traverse to a point P is made along geodesic s
through A, B, . . . . The corresponding distances and angles, as measured o n
the spheroid, are then plotted in the plane as shown in Fig . 1 . Let a different
route from O to P follow A', B', . . . . In the plane the end-point P' is different
from P . If the area bounded by the two traverses is small then the gap PP '
is obtained up to terms of a higher order by rotating P an angle around th e
centre of mass of the polygon OAB . . . PP' . . . B'A' with area w . The angl e
w/MN, where M denotes the radius of curvature o f
the meridian and N denotes the other principal radius of curvature . Thi s
theorem is proved in the appendix . Often M and N were multiplied by
7/180 to give the length of a degree . This notion, which has given nam e
e is equal to the excess
to an entire branch of science, "measurement of degrees", was preferred b y
Bugge [1, p .104] .
Zealand is about 90 by 130 km,
is e = 90 130/(6379 . 6393)
M
6379 km and N
6393 km . The exces s
2 .9 . 10 -4 rad . The maximal distance from
4 For instance, the sum of angles covering the entire horizon around the trigonometri c
station "Hellingehøj ved Høje-Taastrup" (denoted H by Bugge in triangles Nos . 3, 4 an d
6 [1, p .64]) is 10" less than 360° .
86
L.K. Kristensen
MfM 46 : 2
o
Figure 1 : On a spheroid, with principal radii of curvature M and N, we follow a
polygonal line from O to P through A, B, . . . and a different one through A', B ' , . . . .
The figure shows the tracks plotted in the plane with the conservation of angle s
and distances . The end-points P and P' will not coincide . The area of the close d
polygon OAB . .PP' . .B'A'O is w and the excess is E = w/MN . The centre of mass
of the polygon is denoted by C . The gap PP ' is the excess E times the distanc e
PC and is orthogonal to PC in the approximation (A .6) .
the centre to an edge is 79 km so the gap (PP') becomes 2 .9 . 10 -4 • 7900 0
m = 23 m . This corresponds to 1/5 mm on the final map which is hardly
visible .
The example above is an extreme one and the excess E will also be reduced
because the sums of the angles in the individual triangles are adjusted t o
exactly 180° . Even today mathematical ambiguities are justified in practic e
if the errors are negligible for the purpose in mind or hidden by observational
errors . This is for instance the case for the present cadastral coordinates i n
"System 1934" .
The result of the computations in the plane was two coordinates : the dis tance x to the normal orthogonal to the meridian through the Round Towe r
and the distance y between the foot point and the Round Tower along this
MfM 46 :2
87
"perpendicular" . These coordinates are not defined uniquely when computed in the plane but depend on the chosen path . In view of the observational errors these inconsistencies will hardly be noticed if the differen t
paths cover small areas . The coordinates were used as rectangular coordinates in the plane and the draft maps adjusted to these points . Although
the concept of a map projection was not mentioned this implies a projectio n
of the Cassini-Soldner type .
The Earth could not be considered flat, however, in the computation o f
geographic latitude y) and longitude A from the plane coordinates (x, y) .
Referring to Bouguer, Bugge used a non-elliptical spheroid with flattening
1 :179 for the Earth . Its figure is defined by a table of the length (M) of a
(N) as a function of cp . 5 The adopted
geographical position for the origin (x, y) = (0, 0) at the Round Tower wa s
cpo = 55°40'56", ao = 0° .
meridian degree and of the normal
To derive ((p, a) from (x, y) is today a standard problem in geodesy an d
solved by an expansion in e 2 , where e is the eccentricity of the ellipsoid .
When the Earth is not an ellipsoid we must integrate the differential equation for the geodesic curve with arbitrary M and N . This requires a shar p
definition of the rectangular coordinates not given but implicitly used by Bugge himself.
Expanding in terms of the small quantities
M' = dM/dcp,
x/M, y/N, N' = dN/dy,
and
we have to third order :
(P = Po + m+
X
2N
y 2 tan cpo
xy 2 tan 2 cpo
2MN
2MN 2
M'
(2)
and to second order :
A _ A o +
y
(N + 2
MN ') cos(cpo + M)
(3 )
If r denotes the distance along the geodesic and the polar angle cr denote s
the angle between the geodesic at the origin and the meridian, we hav e
x = r cos a and
y = r sin a .
(4 )
5 In modern notation : M = p + q sin4 cp, N = p + q(1 - 2/3 cos 2 + 1/5 cos4 (p) ; p =
58756 .14 favn, q = 992 .86 favn . N > M . The figure of the Earth was defined by M
as given by Bouguer (1698-1758) [1, p .105] . N is derived from M and the flattening is :
5q/(15p + 8q)
1 : 179 .1 . The numerical values of p and q are here adjusted to Bugge' s
table [1, p .104] .
88
L.K. Kristensen
MfM 46 : 2
Formulae (2), (3) and (4) thus give a parameter representation of the
geodesic in terms of the distance r .
Such formulae, or similar ones, were not derived by Bugge . It was not hi s
style and he says explicitly that he did not "fill the book with theorems an d
formulae, which after my conviction, and long experience, are of no use an d
cannot be utilized in practice" [4, p .VII] . He always solved the problems by
some kind of reasoning and geometric intuition .
The first term in (2), the distance x divided by the length of a meridia n
degree (M), seems obvious . The intuition does not, however, indicate tha t
M should refer to the mean latitude . The next, y 2 -term, is the distanc e
between the perpendicular and the latitude circle and is given by a tabl e
with 12 entries [1, p .106] . This table is not smooth, the second difference s
have a random variation of the order +2" . It was probably computed fro m
12 right-angle triangles on a sphere with 6-figure logarithms and given, i n
Bugge ' s habit, to only 1 " . The angle y tan coo/N between the meridian an d
y-axis (meridian convergence) increases with y . The cosine of this angle
must be applied on the main term to project it onto the axis . This gives the
last third order term which is neither estimated nor mentioned by Bugge .
An example [1, p .106] gives the details of the computation of the latitud e
difference between Copenhagen and the Lighthouse at Kullen . The mean
latitude is 56 .0° but the M used corresponds to 57.0° which is outside th e
range of latitudes . This is only one of several examples where Bugge did no t
interpolate but used tabular values, and not even with the nearest argument .
The computation of the longitude A was not performed by the simple formula (3) but by a rather unwieldy method . First r and a were computed
from (x, y) by (4) and hereby defining the precise meaning of these coordinates . The length (R) of 1° of the oblique arc with polar angle a was then
computed by an approximation to Euler's formul a
1 _ sine
R N
a
cos t a
+
M
(5)
Now let P = (0,0) be the origin at the Round Tower, S = (x, y) be the
station in question and Z the North Pole . Bugge now imagined a spherica l
triangle PZS with sides SZ = 90° - co, PZ = 90° - coo and PS = s = r/R.
The longitude is then computed as A = ZPZS . The problem is, however ,
that there is no such spherical triangle because the normals at P and S
do not intersect the Earth axis in the same point due to the non-spherica l
MfM 46 :2
89
spheroid and are not in the same plane . Nevertheless, the procedure give s
some approximation to (3) . The triangle PZS gives in modern analyti c
notation :
2(Po sin 2
cos
~2
4
s
2
-
M2 ~
2
= sine
cos
2
1 (r 2 (si~a
2
sin )2
M a J1 - ~
+ co
2
2
x
Y
sin2 a =
4 RN + 4M ( N NI )
2 Po
/
4N2
(6 )
2
(1 - cos 2
N
\ 2)
a \1 _ M
The first line in (6) is derived from the spherical cosine-relation and i s
the exact form of the familiar approximation s 2 = (cp - (po) 2 + (A cos cp) 2 .
Expanding in small quantities and inserting (4) and (5) gives
reductions - the final term .
after some
For M N this is an approximation to (3) but in the limit r -* 0 it doe s
not converge correctly to this expression . The method does not represent a
lasting solution to a main problem in geodesy .
An example with y = 66850 favn, cp = 55°01'34" and N = 59568 is give n
in detail in [1, p .110] . Bugge's inconvenient method gives 1°57'29" whil e
formula (3) simply gives :
66850/(59568 x cos55 °01'34") = 1°57'28"
(7)
We note that N refers to the mean latitude but the argument of the cosin e
in (3) refers to the actual latitude - a fact not evident by intuition . Again ,
in this example Bugge does not use the mean latitude (55 .4°) but 56 .0 °
outside the range .
Evidently Bugge did not waste time over such trifles . His goal was th e
printed maps and for economy and speed the tolerances were adjusted to thi s
purpose . Bugge actually achieved his goal and the field work was finishe d
in 1821, a few years after his death .
3 Caspar Wesse l
From the very beginning it was the duty of every surveyor also to trai n
apprentices in the field . One of Bugge's first apprentices was an elder brothe r
90
L.K. Kristensen
MfM 46 : 2
of Caspar Wessel . From 1764 Wessel was himself involved in the work as his
brother's assistant .
We have seen above the kind of theory and the standard of applied mathematics which met Wessel . There seems to be little stimulation for a gifte d
mathematician . There was no "higher mathematics" in the sense of analysis, differential equations, series expansions or so forth . Such topics wer e
not a part of Bugge's curriculum - on the contrary! In his text-book [3 ,
p .15] Bugge criticized the physicist H.C . Ørsted for using differential calculus as being both "superficial" and "ill-timed", because the problem in
question could be clearly explained in more familiar terms . 6 Bugge's mai n
tools remained the 6-7 figure logarithmic tables and trigonometry.
Wessel surveyed in the field during the summer and compiled the maps i n
1 :120000 from originals in scale 1 :20000 during the winter . The first map ,
with his name on it, was finished in 1768 and covered NE Zealand . In rapi d
succession appeared the three maps covering the remaining parts of Zealand .
Finally, in 1777, the four maps were put together to a single map in 1 :240000 .
In 1779 Wessel was appointed trigonometrical observer with the duty t o
push the triangles further west in order to cover the whole of Jutland t o
the Elbe, as Slesvig and Holstein were in those days in personal union wit h
Denmark . When Bugge wrote his book [1] others had succeeded him in th e
field . The westernmost trigonometric station reached was denoted "Heslebierg" (nowadays "Hesselbjerg") in Jutland 3°11 .5' west and 1 .6' south o f
Copenhagen . Bugge erroneously placed this station on the island of Fune n
[1, p .113] .
When Wessel took over and started from there he proposed a plan for th e
continuation of the work which was approved by The Royal Danish Academ y
of Sciences and Letters at a meeting on February 5 1779 [13, p .193] . Thi s
plan for the division of the country is often mentioned, among others, in a
biography [16], but its content is not known [11, p .106] . Let us now try t o
reconstruct this plan from the distinct appearances of the published map s
before and after Wessel was in charge of the triangulation .
6 The university professor criticizing the work of a colleague in his text-books mus t
have amused the students . Bugge also warns against using Ørsted's value for the lengt h
of the new French metre in terms of the Danish unit fod [3 . p .132] . After his death, Bugge
was himself criticized by the influential Ørsted brothers ; the physicist's brother became
prime minister .
MfM 46 :2
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J
F
/II4IIII(
12i
.f llIIIII
l~
1111
n
75
sos
//
Jag ,--ffi e,
iMlmlrtnlrt 4 .,5
Figure 2 : The copperplate of Thomas Bugge's textbook (ref . 2, copper XII) shows
the outline of his concept of a map . The central meridian is AB, to which th e
borders ED and FG are parallel and the North and South borders FE and CD
are orthogonal . Due to the meridian convergence such rectangular frames cannot
be joined together with neighbouring sheets at their edges .
92
L.K. Kristensen
MfM 46 : 2
Bugge's maps had a rectangular shape which was symmetric relative to a
central meridian (AB in Fig .2) . The central meridian and the two latitud e
parallels (CD and EF), which were practically straight lines, were in tru e
scale (Flamsteed projection) . Due to the meridian convergence the uppe r
boundary covers 1 .1' more in longitude than the lower . A main point is tha t
the sheets could not be joined together at the border lines due to an overla p
and the non-parallel rectangular frames . The centre of Zealand, as covere d
by the four maps, is shown in Fig .3 . It appears very confusing with a smal l
area having as much as three times overlap [1, p .128[ .
55°40'
2
1
1
- ..TE
2
~SE
3
2
SW '
NW
55°30 '
2
1
1
55°20'
1 °00'
0°50'
0°40 '
0°30'
0°20 '
Figure 3 : Zealand is covered by 4 maps (NE, NW, SE and SW) and the figure show s
how they overlap . The numbers of overlaps (max . 3) are indicated . The maps ar e
also slightly twisted due to meridian convergence . The appearance is confusing bu t
Thomas Bugge made the best use of the given paper size ("Grand Aigle") and
the scale 1 :120000 . Caspar Wessel's frames were equal rectangles (at first 131- by
71, later 131 by 8 mile ) fitted together on both sides of the meridian 3°12 ' W o f
Copenhagen as a common border .
MfM 46 :2
93
The new maps were divided by a rational system, and Bugge mentions [1 ,
p .VI] that it was decided that the future maps could be joined together .
Inspection of the published maps shows that their east or west border s
are 3°12' west of Copenhagen . This is exactly the meridian through th e
trigonometric station Hesselbjerg mentioned above . This division is als o
alluded to several times in the literature [11, p .67,107,131] . The north an d
south borders are then not latitude parallels but orthogonals to the 3°12 '
W meridian . The latitude scales of the east and west borders are then not
identical but displaced by the y 2 -term in (2), or 0 .66' .
This rational system was probably invented by Wessel in 1779 and ever sinc e
rectangular maps, which could be joined, were used in Denmark . This ide a
may have been Wessel's main lasting contribution to cartography . His actual
field work was necessarily surpassed . However, the production of maps which
cannot be joined is continued elsewhere, for instance the Prussian polyede r
projection and the International Map of the World .
In 1805, bowed down by age (59 years!) and feebleness Wessel applied fo r
retirement with full salary [13, p .199] . His work had always had a hig h
reputation and when in 1808 the French, then allied to Denmark, asked fo r
the survey of Slesvig and Holstein Wessel undertook the copying . Actually
the French only received the Holstein data . By a diplomatic excuse the
king avoided forwarding a copy of the Slesvig survey! Wessel renounced th e
pay and received a silver-medal, books and maps instead [13, p .413] . The
intention of Napoleon's France was to have maps from Trieste to Slesvi g
[11, p .156] .
4 Developments after Bugge and Wesse l
During the last part of Bugge's and Wessel's active working periods ther e
was an increasing demand for better maps . All over Europe the enclosure
movements in agriculture demanded large scale cadastral maps and th e
armies wanted relief maps with more topographical details .
Napoleon once said that a good map was half the victory. In 1808 he planne d
a new atlas of France, in scale 1 :80000, to replace the old Cassini maps .
The work on this "Carte de 1'Etat Major" started after his fall in 1815 an d
continued until 1880 . It was using Bonne's projection (after Rigobert Bonne
(1727-95), in Denmark denoted : "modified Flamsteed projection") with the
94
L .K. Kristensen
MfM 46 : 2
central meridian through the Paris observatory . The southern latitude (45° )
of osculation reflects that Italy and Spain were within Napoleon's sphere o f
interest . Occupied German-speaking areas, like Hanover, were surveyed b y
the French army . As we have seen, Wessel provided similar data for Holstein .
Already during Bugge's last years, plans were made by the Danish Genera l
Staff to prepare improved maps based on the original measurements of th e
Royal Academy, [9, p .24] . In 1842 the Academy stopped the productio n
of maps and transferred the archives to the General Staff which had no w
planned to produce an atlas in 1 :80000 based on a quite new triangulation .
Like the French, the projection should be Bonne's with the central meridia n
2°12' west of the Round Tower and osculation at 56°N . The relief was finall y
shown by equidistant level curves .
The French map in 1 :80000 did not stand the test of the First World War .
The trenches were so close that they could not be shown with accurac y
on any scale smaller than 1 :20000 so large scale maps were redrawn [14 ,
p .255,261] . The generalization errors inherent in the signatures became to o
large for the artillery and the British expeditionary force had to re-surve y
their entire front lines . For this work a conformal projection would have
been an advantage . The Bonne projection, however, has a shear
( cP - Po) x
(a -
Aa) x cos cpo
(8)
which amounts to 16' in the disputed area of western Germany and which
was inadmissible in computations . According to Deetz : "In the rigorou s
tests of the military operations these errors became too serious for the pur poses which the maps were intended to serve" [6, p .71] . The conical conformal Lambert projection was used in a later French survey and by learnin g
this the Americans became enthusiastic supporters of conformal projections :
"the excellent qualities of the Lambert projection, although not unknown ,
had been systematically forgotten for some time and were evidently no t
fully appreciated until the beginning of the First World War" [6, p .213,90] .
However, in this respect the Danish survey had been far ahead . The theor y
of the Lambert projection on an ellipsoid appeared in 1868 [14] . About thi s
time the survey by the General Staff was finished on the Danish island s
and was about to start in Jutland . As previously this was taken as a n
opportunity for improvements and the conformal Lambert projection wa s
introduced .
MfM 46 :2
95
5 Conclusion
The geodetic work of Bugge and Wessel was destined to become obsolet e
already during their lifetime . The revolution of geodesy was then alread y
in the mind of C . F . Gauss . In 1799 a letter [7, p .136] by Gauss to th e
editor of "Allgemeine Geographische Ephemeriden" (F . von Zach) treate d
the determination of the figure of the Earth by several arcs of the meridian combined by the method of least squares . Exactly the same data wer e
simultaneously analysed by Bugge [5, p .586-588] . In a letter to Schumacher
[8, p .345] from 1816 Gauss offered to compute the Danish first order tri angles by his own (read : least squares methods) . In the same letter he also
proposed the problem of conformal projections for a prize essay
which h e
won himself in 1825 . Incidentally he mentioned in the letter the great ad vantage that Denmark had already been surveyed once - but an advantag e
only in connection with the reconnaissance for trigonometric stations! H e
complained about this time-consuming work and that he could not obtai n
similar data from the French survey of Hanover .
There is a vast contrast between the good workmanship of Bugge and Wes sel, which we have seen above, and the mathematical genius of Gauss . The
importance of the work of the latter can only be put in the proper perspective by a comparison with that of his predecessors . We have seen how
badly Bugge needed a statistical theory of errors and he explained [5, p .580 ]
how the repartition of specific errors was considered quite arbitrary an d
was a matter of debate between the members of the International Commission of the Metric System . Gauss practically removed this arbitrariness an d
introduced rigour in the entire theory .
6 Appendix : Proof of the gap theore m
Bugge's triangles were computed in the plane . This was done at that tim e
also in France and some German states . We shall here estimate the approximation of this procedure . The first rigorous definition of rectangula r
coordinates on the sphere was introduced by J . von Soldner (1776-1833) i n
1809 in a survey of Bavaria . Let y, counted positive towards the east, be th e
shortest distance AA' from a given point A to its projection A' on a chose n
prime meridian and x the distance from A' to a fixed point on the meridian
96
L .K. Kristensen
MfM 46 : 2
measured positive towards the north . ? From the point
A = (x, y) we now
go to B by a distance s to a point B along a geodesic that at A makes an
angle a with a curve equidistant to the prime meridian . Reference 10, p .26 5
(modified here by a generalization to a spheroid, essentially a substitutio n
of
r .2 by MN) gives for the Soldner coordinates of B = (x', y') :
x' = x+scosa+sy t cosa/2MN,
y ' = y+ssina
(A .1 )
and the direction at the end-point
a ' = a - sy cos a/MN.
(A .2)
If the same distance s is drawn as a straight line in the plane the endpoin t
will have the coordinate s
x ' = x +scosa,
y ' = y +ssin a
a will not be changed at B. We now consider AB as bein g
a part of the traverse GAB . . . P . By stretching s we have displaced the
remaining part of the traverse by (sy2 cos a/2MN, 0) and we have rotate d
it by an angle sy cos a/MN . This will displace the endpoint P = (xp, yp ) by
and the angle
(sy 2 cos
a/2MN, 0) + sy cos a/MN • ((yp - y), -(xp - x))
(A .3)
Thus if we replace the entire geodesic polygon by its plane counterpart w e
can obtain the total displacement of P by integrating the expression (A .3 )
along the polygonal path GAB . . .P. Setting dx = s cos a we get :
.P
1
MN
MN
fP
JO
dx ((yyp - y2 / 2) ; - y (xp - x ))
dx
J p dy ((y - y ) ,-(x - x)) .
(A .4)
// o
If we subtract the corresponding integral for the alternative route OA'B '
P we obtain a double integral bounded by the closed figure with are a
w:
MN
fdxfdY(yp_y),_(x p_ x)) .
7 Note: Soldner's coordinates form a left-hand system .
(A .5 )
MfM 46 :2
97
Introducing the centre of mass coordinates (x e , ye) this integral is easily
evaluated to the rotation s around the centre of mass :
F ((yp -
where s =
w/MN is
y~} -( x p - xe)),
(A .6 )
the excess .
7 Addendum
At the Wessel Symposium I learned that Bodil Branner and Nils Voje Johansen had procured Wessel's original handwritten reports from the archive s
of "Kort- og Matrikel Styrelsen" . The notes for 1787 describe the plan fro m
1779 mentioned above . What was concluded above from the published map s
is essentially confirmed . The puzzling increase of the rectangular frames o f
the maps from 132 x 72 mi l 2 to 132 x 8 mil 2 is, however, explained in Wessel' s
notes . The country extended farther north than expected in advance of the
survey! The spheroid with flattening 1 :179 used by Bugge was replaced by
an ellipsoid with flattening 1 :230 . This ellipsoid, based on older pendulu m
observations by Richer, is mentioned by Bugge [2, p .284] . Wessel ' s concept s
of map projections do not seem clarified . It seems that he computed rectangular coordinates by zigzagging along narrow strips on the ellipsoid an d
then unfolded the strips in the plane . The computations were made by imaginary numbers and took into account the meridian convergence ignored by
Bugge . The method was affected by the ambiguities described by the abov e
"gap theorem" .
8 Acknowledgement
The author is much indebted to Kurt Niøller Pedersen for drawing attentio n
to the present interest in the mapping by the Royal Academy and the Wesse l
Symposium and is also much indebted to Anette Skovgaard for preparin g
the electronic version of the manuscript .
Reference s
[0]
Andersen, Einar : Thomas Bugge . Geodætisk Instituts Forlag . Copenhagen 1968 .
98
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
L .K. Kristensen
MfM 46 : 2
Bugge ; Thomas : Beskrivelse over den Opmaalingsmaade, som er brugt ved de
Danske geografiske Karter . Copenhagen 1779 .
Bugge, Thomas : De første Grunde til den sphæriske og theoretiske Astronomie, samt
den mathematiske Geographie . Copenhagen 1796 .
Bugge, Thomas : De første Grunde til den rene eller abstrakte Mathematik. 1 . del .
Algebra . Copenhagen 1813 .
Bugge, Thomas : De første Grunde til den rene eller abstrakte Mathematik. 3 . del .
Landmaaling . 2 .ed . Copenhagen 1814 .
Bugge, Thomas: Thomas Bugge's Rejse til Paris i Aarene 1798 og 1799. Copenhagen 1800 . Partly translated in : M . P . Crosland : Science in France in the revolutionary Era. Cambridge, Massachusetts 1969 .
Deetz, C . H ., Adams, O . S . : Elements of Map Projection . U .S . Department of Commerce, Coast and Geodetic Survey . Special Publication No . 68, 5 . ed . Washington
1945 .
Gauss, Carl Friedrich : Werke VIII. B . G . Teubner, Leipzig 1900 .
Gauss, Carl Friedrich : Werke XI/2 . Julius Springer, Berlin 1924 .
Henriksen, P. G . : Hærkort i Danmark og nabolande gennem tiderne. Geodætisk
Institut . Copenhagen 1971 .
Jordan, W . : Handbuch der Vermessungskunde 3 . 3 . ed . Stuttgart 1890 .
Lomholt, A . : Kgl . danske Videnskabernes Selskabs historie 1742-1942. Bd .IV .
Copenhagen 1961 .
Maire, A . L . le : Generalstabens Kaartlægning af Danmark . Militært Tidsskrift X,
1880-1881 . Copenhagen 1881 .
Molbech, E . : Det kgl . danske Videnskabernes Selskabs Historie 1742-1842. Copenhagen 1843 .
Winterbotham, H . S . L . : British Survey on the Western Front . Geographical Journa l
53 (1919) 253-27 6
Wittstein, T . L . : Ueber conforme Karten-Projectionen . Astron . Nachr. 71 (1868 )
369 .
Zeuthen, H . G . : Wessel, Caspar : Dansk Biografisk Leksikon . Bd . 6 . Copenhagen
1904 .
MfM 46 :2
99
Heinrich Christian Schumacher mediator
between Denmark and Germany ; Centre of
Scientific Communication in Astronom y
Jürgen Hamel *
1 Communication in Astronomy
The structure of scientific communication in the field of astronomy ha s
undergone manifold changes during the past 500 years . The main factor s
determining this process are the following :
1.
The number of persons interested in scientific communication ,
2.
The amount of material suitable for scientific communication ,
3.
The traffic situation for scientific communication as far as speed, security and cost are concerned .
For centuries two forms of communication have been of great importance :
1.
correspondence, an d
2.
academic lectures passing new points of view about scientific fact s
from master to student . Since important universities recruited students from a wide geographical area, and since students used to mi grate from university to university, hand written lecture notes circulated among them, providing thereby efficient structures of communication .
The exchange of letters was of less importance . As an extraordinary exampl e
in this respect the letters exchanged between Johannes Regiomontan an d
Giovanni Bianchini in 1463 should be mentioned as a game of questions an d
* Archenhold-Sternwarte, Alt-Treptow 1, D-12437 Berlin, Germany
100
J. Hamel
MfM 46 : 2
answers measuring mental and creative powers . 120 years later a historically
very important correspondence between Wilhelm IV, landgrave of Hessen
in Kassel, and his astronomer Christoph Rothmann with Tycho Brahe too k
place .' Supported by Wilhelm, Tycho Brahe on May 23, 1576 had bee n
granted the island of Hveen by his feudal lord the Danish King Frederi k
II in addition to 400 thalers for the construction of an observatory . 2 Th e
letters exchanged between Hveen and Kassel between 1585 and 1590 wer e
apparently delivered by Brahe's disciples when they travelled back and fort h
from the book fair in Frankfurt . They are still an important source for the
history of astronomy.
Nevertheless the need for scientific communication was still very limited ,
except on special occasions such as the introduction of the Gregorian calendar in 1582 and the subsequent discussions in the Protestant countries ,
as well as the appearance of comets, eclipses or other celestial phenomena .
On such occasions opinions and ideas were exchanged by letters or small
printed volumes of high circulation and low price .
A hundred years later, by the end of the 18th century, firm structures of
research and communication in the field of astronomy had been established .
There existed a number of scientific research institutions of internationa l
importance and reputation . Among the earliest were the observatories in
Greenwich, Paris, Copenhagen and, a bit later and as such rather modest ,
in Berlin . In Copenhagen an observatory was set up already under Brahe ' s
student Christian Severin Longomontanus, author of the recognised manual
Astronomia Danica (1622) . This observatory became really famous unde r
Ole Rømer, who had been able to measure the speed of light very precisely.
He constructed and used new instruments, as for instance the prototype s
of circle instruments which after 1750 led to the great success in celestia l
mechanics .
About 1750 astronomy with its modern structure was represented at univer sities and academies . Since practical and theoretical astronomy was practised in many places there was a great demand for higher quality in the
exchange of observations and their mathematical treatment . In addition ,
'Hamel, Jürgen: Die astronomischen Forschungen in Kassel unter Wilhelm IV. Mi t
einer Teiledition der deutschen Ubersetzung des Hauptwerkes von Copernicus um 1586 .
Thun ; Frankfurt a . Main 1988 (Acta Historica Astronomiae ; 2 )
2 Dreyer, J .L .E . : Tycho Brahe . Ein Bild wissenschaftlichen Lebens im sechszehnte n
Jahrhundert. Karlsruhe 1894 ; Reprint Vaduz 1992, 91
MfM 46 :2
Schumacher mediator between Denmark and Germany
10 1
the changed social demand for astronomy had to be taken into consideration . It was no longer astrology, and casting of horoscopes, that brought
high prestige to astronomy . The interest predominant came from the determination of the latitude and longitude at sea, surveying and improvin g
the calendar . In addition, astronomy gained public interest due to Newton' s
physics, the very precise methods of orbit calculation and the new result s
in comet research, reflected in ideological, philosophical and theological systems .
As long as astronomical research was rather sporadic and the individua l
scientists did not have to'discuss and agree with each other very much, th e
number of persons interested in communication had remained very small .
This was now changing profoundly . Correspondence as a means of exchan g
information started to play an important role ; membership in a circle o f
corespondents was of great importance as a pre-condition for one's ow n
scientific research as well as for propagating it . Of course, all this went
alongside the publications of the academies, independent printed matter an d
the first scientific journals, often founded in relation to scientific academies .
For the German language area the yearly Ephemeriden by Gottfried Kirch ,
published since 1681, should be mentioned . They had a literary annex where
Kirch used to publish research of his own as well as results which he learne d
about in letters from his correspondents or in new books . Kirch continue d
these literary annexes in his calendars published by the Berlin Academ y
after 1700 and the tradition was carried on by Johann Heinrich Lamber t
and Johann Elert Bode in the Berliner Astronomisches Jahrbuch founde d
in 1774 . 3
The publication of an astronomical annual supplied a demand among astronomers . As a proof one should mention the great number of contribution s
that flooded Bode, who could hardly manage the contributions . Almost all
contemporary astronomers were among his authors . He usually publishe d
N.N. to the publisher" .
Nevertheless, a real handicap was the lack of a fast outlet for informatio n
due to the annual publication only.
their writings in the form of "From a letter from Mr .
3 Hamel, Jürgen : Geschichte der Astronomie . Von den Anfängen bis zur Gegenwart . Basel 1998, 271 ; Herrmann, Dieter B . : Die Entstehung der astronomischen Fachzeitschriften in Deutschland (1798-1821) . Berlin-Treptow 1972 (Archenhold- Sternwarte /
Veröff. ; 5), 15-74
102
J. Hamel
MfM 46 : 2
This shortcoming was made up by Franz Xaver von Zach, whose work was
actively supported by the Duke Ernst II of Sachsen-Gotha . In 1798 Zac h
founded an astronomical-geographical journal, the monthly Allgemeine Ge-
ographische Ephemeriden. The combination of astronomy, cartography an d
geography, however, turned out to be a failure very soon . Despite all the
connections between these two disciplines mainly through the geographica l
determination of latitude and longitude, they had progressed in their specialisation to such an extent that they had to be covered by special journals .
Therefore, in 1800 Zach founded the Monatliche Correspondenz zur Beförderung der Erd- and Himmelskunde, a specialist journal of astronomy
where geography was included mainly in so far as it was relevant for th e
geographical determination of latitude and longitude . With a change when
Bernhard August von Lindenau became the publisher the Monatliche Cor-
respondenz existed till 1814 as the central means of communication betwee n
the astronomers next to the literary annex of Bode's annual . It was discontinued because Zach went to France to found a new journal, the Correspondance Astronomique, which was published from 1818 . In the beginnin g
it contained papers from well-known astronomers, among them the youn g
Heinrich Christian Schumacher . However, the journal could not gain real
importance . It published articles in French and Italian only, Genua, wher e
it was published was too far from the centres of astronomy, and the postal
service was insecure due to political turmoil . Publisher Zach soon becam e
isolated and in 1826 he stopped its publication .
At the same time Lindenau together with Johann Gottlieb Friedrich v .
Bohnenberger published the Zeitschrift für Astronomie and verwandte Wissenschaften, with a circulation of 200 copies and strongly concentrated o n
astronomy. They received contributions from the most important scientists ,
but the two co-publishers did not collaborate very well . Moreover, Lindenau
found himself in a difficult situation, being a state employee of Saxony wit h
manifold political duties (he later became state minister of Saxony) . The
journal was discontinued in 1818, only three years after it was founded .
These detailed introductory remarks are necessary in order to understan d
the problems of scientific communication and the difficult problems of th e
foundation and maintenance of an astronomical journal . There was the need
and demand, and whenever a publication was stopped it was regretted ver y
much because the astronomers lost the possibility to publish their results o f
research . But nobody was ready to undertake the troublesome efforts of a
MfM 46 :2
Schumacher
mediator between Denmark and Germany
10 3
publisher - neither Gauß nor Olbers, Encke or Bessel .
2 Heinrich Christian Schumache r
Now the time has come to turn to Heinrich Christian Schumacher . 4 Schumacher was born in 1780 in Bramstedt/Holstein near Hamburg as a so n
of the chamberlain and Danish senior civil servant in Segeberg Andrea s
Schumacher . His father had apparently a close relation to the King . Indeed ,
Heinrich Christian was introduced to King Frederik VI at the age of 7 . 5
This acquaintance turned out to be important both for Schumacher and fo r
astronomy, in so far as the well-educated King till his death protected an d
promoted Schumacher in many ways .
After Schumacher had lost his father at the age of ten, he was educate d
by the erudite pastor J .F .A . Dörfer who was known for his topography of
Schleswig and Holstein . Later he attended the grammar school in Altona ,
whose headmaster J . Struve became the progenitor of a well-known family
of astronomers . From 1799 he studied jurisprudence in Kiel and Göttingen
and in 1804 he went to Livonia as a private tutor and soon became a lecture r
of law in Dorpat . Simultaneously he studied mathematics and astronom y
together with Johann Wilhelm Andreas Pfaff. His doctoral thesis in 180 6
dealt with Roman Law .
Some remarks concerning the history of Holstein will not be amiss . Traditionally Holstein as a county, later duchy had its cultural and political
position between Denmark and the German lands . Belonging to the German Reich on the one hand, it was under Danish fief on the other . At the
Vienna Congress it came under the rule of Denmark and the Danish Kin g
became the Duke of Holstein . At the same time Holstein belonged to the
loose German Alliance as for instance Hanover, too, politically belonging t o
the English Crown .
4 Olufsen, Christian Friis R.ottböll: Heinrich Christian Schumacher . Oversigt over det
Kgl . danske Videnskabernes Selskabs Forhandlinger og dets Medlemmers Arbeider i Aare t
1851 . København, 226-235, German transi . : Astronomische Nachrichten 36 (1853), 393404 ; Petersen, Adolph Cornelius : Schumacher's Tod . Astronomische Nachrichten 3 1
(1851), 369-370 ; Repsold, Joh . A . : FLC . Schumacher . Astronomische Nachrichten 20 8
(1918), 17-34
5 Briefwechsel zwischen C .F . Gauss und H .C . Schumacher . Hrsg . von C .A .F . Peters ,
vol . 3 . Altona 1860, 345
104
J. Hamel
MfM 46 : 2
Figure 1 : Heinrich Christian Schumacher (Nationalhistoriske Museum, Frede-
riksborg )
This status quo was practised peacefully in Holstein . The Danish King respected the German language and culture and the population accepted th e
Danish rule . By birth Schumacher was a Danish subject, he felt himself a s
such and turned this circumstance to his own advantage and to the advantage of astronomy.
MfM 46 :2
Schumacher - mediator between Denmark and Germany 10 5
A promised employment at the King's chamber of finance came to nothin g
due to war accidents in Copenhagen, but in 1810 Schumacher was offered a n
extraordinary professorship of astronomy at the University of Copenhagen .
At the same time he made the acquaintance of important astronomers an d
constructors of instruments such as Johann Georg Repsold in Hamburg an d
exchanged letters with Friedrich Wilhelm Bessel in Königsberg, Johann E lert Bode in Berlin and Wilhelm Gibers in Bremen . Having been granted a
King's scholarship , 6 see figure 2), he worked in 1808/09 for a year with Gauß
in Göttingen . In 1812 he got a leave of absence and became the director o f
the observatory in Mannheim because of disagreement with the professor of
astronomy Thomas Bugge . This appointment must have been offered on the
initiative of Bernhard August von Lindenau, who not only directly patronised him, but also personally supported this decision : "Surely you are absolutely right to prefer the friendly Mannheim with its excellent instrument s
to the harsh Copenhagen with, as I think, its bad instruments" . 7 However ,
to his Danish compatriot Hans Christian Ørsted, secretary of the Royal Danish Academy of Sciences and Letters since 1815, Schumacher complaine d
that his position in Mannheim was a bit difficult because the observator y
was rather modestly equipped with instruments and " in an incomprehensible state of neglect and disorder" . 8 But he did not get the time to modernise
it because already in 1815, after the death of Bugge, he returned as ordinar y
professor of astronomy and director of the observatory in Copenhagen . O n
December 8 of that year the Royal Danish Academy of Sciences and Letter s
admitted him as a member . 9
It was his desire to return to Denmark even with financial loss, becaus e
obviously Schumacher did not feel well in Mannheim, as he had told Ørste d
before . "My dearest friend, maybe you could remind the King of me?", h e
6 Rigsarkivet, København . Fonden ad usus publicos 1810-1812, 1808 Apr . 26 ; Fonden
ad usus publicos . Aktenmaessige bidrag til belysning af dens virksomhed . Udgivet a f
Rigsarkivet, vol . 2 . København 1902, 123-12 4
' Schumacher-Nachlaß, Staatsbibl . Berlin, Preuß . Kulturbesitz, Lindenau 1, 15 .2 .181 3
( " Gewiss haben Sie in jeder Hinsicht Recht, das freundliche 1Vlannheim, mit seinen vortrefflichen Instrumenten, dem rauhen Copenhagen mit den dortigen wie ich glaube schlechte n
Instrumenten vorzuziehen ." )
$ Ørsted, Hans Christian : Correspondance avec divers savcents. Publ. M .C . Harding, vol .
2 . København 1920, 512 ("Mir wurden die Instrumente in einem unbegreiflichen Zustand e
der Verwahrlosung und Unordnung übergeben ." )
9 Det Kongelige Danske Videnskabernes Selskab 1742-1942, Samlinger til Selskabet s
Historie (ed. A . Lomholt), vol . 1 . Kobenhavn 1942, 366
106
J. Hame l
MfM 46 : 2
Figure 2 : Schumacher's application for a scholarship to visit foreign observatories ,
Altona 26 .4 .1808, 1 (Rigsarkivet, København . Fonden ad usus publicos 1810-1812 ,
for the full text see the appendix)
MfM 46 :2
Schumacher mediator between Denmark and Germany 10 7
asked Ørsted pleadingly. In fact immediately after Bugge's death on Januar y
15, 1815 everything was prepared for the return of Schumacher . Already o n
March 2 he could inform Ørsted that he had received "a kind of recall" »
But in the "Round Tower" in Copernhagen he found antiquated instruments . ll In order to take up important scientific work he conceived th e
project of a new Danish land surveying from Skagen to Lauenburg (at that
time belonging to Denmark) . It was to replace the hardly completed surveying done under the auspices of the Academy to which Bugge and Wesse l
had contributed . To get the King's support for the plan he argued in Apri l
of 1816 in the following clever way :
If you would allow me to add a few words concerning my own per son then I would like to say that here [in Copenhagen] I sit aroun d
in such complete idleness as an astronomer can ever do . The observa tory is simply no good for any scientific use and I cannot make an y
observations there of the slightest value . . . However, all my life an d
work is devoted to astronomy and my strongest desire urges me to d o
this work . Having left an excellent observatory in Mannheim, may I
at least hope to find support for scientific work in my homeland, es pecially when it will reflect the major part of Denmark's glory? Eve n
more, should the financial situation after some years make it possibl e
to built an observatory that might serve science, this work would fil l
the time in between . 1 2
One might consider his evaluation of the Mannheim observatory to be a
tactical exaggeration . However, the plans of land surveying were approved ,
and before long an observatory was build in Altona (not in Copenhagen) .
For the project of a new observatory, started in 1818, Schumacher got first class instruments by Repsold in Hamburg and practical support from the
Danish officers Alexander Caroc (1784-1827), Johann David Nehus (1791 1844), Christian Wilhelm Nyegaard (1796-1846) and Christian Christophe r
' °Ørsted, H .C ., Correspondance,
513, 516 ("Könnten Sie vielleicht nicht wertheste r
Freund den König an mich erinnern und ihm den Wunsch äussern Ihren Freund wieder z u
sehen?") ; for Bugge's biography see : Dansk Biografisk Leksikon, vol . 3 . København 1979 ,
59-61, his publications : Fortegnelse over det Kongelige Danske Videnskabernes Selskab
publikationer 1742-1930 . København 1930, 38-40
" Schumacher, H .C . :
Sternbedeckungen in Copenhagen . Astronomische Nachrichten 1
(1823), 195-19 6
12 Nielsen, Axel V . : H .C .
Schumacher and the observatory at Altona during the war o f
1848-50 . Meddelelser fra Ole Rømer-Obs . Århus 22 (1951), Nr . 22, 267-268
108
J.
Hamel
MfM 46 : 2
Zahrtmann (1793-1853), later the minister of the Royal Danish Navy . Schumacher spent the winter of 1820 working in Copenhagen, where the Kin g
had allowed him to construct a small observatory at "Holmens Bastion "
where he could measure latitude and longitude . According to informatio n
from 1837 the King granted 30 000 Reichsbankthaler per year for the lan d
surveying in Holstein . 13 From Copenhagen it was rather troublesome to or der instruments and to collaborate with Gauß . Therefore Schumacher rente d
a few rooms in Altona to which he moved in June of 1821, having been temporarily relieved from his duties and ordered to report regularly to the King .
This was a very favourite situation for Schumacher . He was still officiall y
professor of astronomy in Copenhagen and therefore financially safe, and a t
the same time he could deal with his extensive scientific work undisturbe d
by any hustle and bustle at the university.
In the garden of his roomy house that he bought in 1821 with the King' s
support at the "Palmaille" in a noble living quarter from 1800 Schumache r
built a little practical observatory . Often he welcomed prominent astronome r
colleagues there, among them Bessel and Gauß . Later he bought an additional neighbouring house, again with financial help from Copenhagen . 1 4
From his studios he could overlook the river Elbe .
The land surveying, the elaboration of a map of Holstein, the revision of th e
system of Danish weights and measures and its comparison to the Prussia n
and French ones occupied his time for many years . But this is not the subjec t
of the present paper . Still let me make two comments : To give reasons fo r
the Danish measurements of latitude and longitude Schumacher referre d
to the resulting possibility of fixing the shape of the earth . The forme r
French and English measurements, he argued on September 14, 1816, ha d
led to contradictory results . Therefore Denmark with its own activities i n
this field could bring the decision . 15 In fact, in 1840 Bessel used the Danish
measurements of latitude and longitude for his very exact calculation of th e
flatting of the earth .
In his own works Schumacher, of course, knew about the results of hi s
predecessors, among them Caspar Wessel . In a letter from Ørsted date d
13 Encke-Nachlaß, Archiv der Berlin-Brandenb
. Akademie der Wiss . Berlin, C .H .F . Peters, 18 .7 .183 7
14
Schumacher-Nachlaß, Mösting 1, June 1821, Jan ./Feb . 182 6
''Rigsarkivet . Fonden, 14 .5 .-14 .9 .1816 ; Fonden ad usus publicos . Aktenmaessige bidrag ,
231-237
MfM 46 :2
Schumacher - mediator between Denmark and Germany
10 9
February 3 1827 one reads that the general staff had ordered him to sen d
"the journals concerning the triangulation of Seeland to the general staff
of the quarter master [Generalquartirmeisterstab] . . . The quarter master
says in his letter that he believes there exist 5 journals by Bugge and 5
by Wessel . He promises to return them very soon ." The General combinin g
several special maps "had found some important inaccuracies" that so far
have not been clarified . In his letter dated February 20 Ørsted confirme d
the receipt from Schumacher . 1 6
Moreover it should mentioned that Schumacher together with Ørsted developed the plan to establish a workshop for precision instruments at the
Polytechnical Institute of Copenhagen to reduce the dependence of Den mark in this field . "Under these circumstances, I am convinced you wil l
have a workshop soon that can compete with any workshops abroad an d
replace them", Schumacher wrote on November 23, 1830 . 1 7
3 Astronomische Nachrichte n
Let us return to the times of the failure of the astronomical journals founded
around 1800 . Since about 1810 Schumacher had been acquainted with th e
most important astronomers of his time . Despite the fact that in scienc e
Schumacher had not achieved much, he was highly regarded, and it was gene rally believed that he would become an efficient astronomer in the future .
"In this Doctor Schumacher, who stayed with Gauß this summer I have
met a very talented and clever man promising a lot for astronomy", Olber s
wrote in November 1809 to Bode in Berlin .' 8 Like many other astronomers
Schumacher worried about the question how to guarantee the existence of a
special astronomical journal . In September 1819 he asked Olbers in Bremen :
"Would it be possible for you to take over the journal of astronomy now
16 Schumacher- Nachlaß, Ørsted 1, 3 .2 ., 20 .2 .1827 [the Schumacher estate in the Staats Berlin contains 64 letters from Ørsted to Schumacher 1817-4850, not mentioned i n
Ørsted's printed Correspondance ]
17 Ørsted, H .C ., Correspondance, 518-519 ; Schumacher-Nachlaß, Ørsted 2-3, 4 .12 . 183013 .9 .1831 ; Ørsted was the director of the Institute since 182 9
18 Olbers, Wilhelm : Aus einem Schreiben . . . v . 21 . Nov . 1809 . Astronomisches Jahrbuch
für das Jahr 1818. Berlin 1810, 256 ("An dem Doct . Schumacher, der sich diesen Somme r
bey Gauß aufgehalten hat, habe ich einen sehr talentvollen und geschickten Mann kennen
gelernt, von dem sich die Sternkunde noch viel zu versprechen hat" .)
bibl.
110
J. Hamel
MfM 46 : 2
that Lindenau has resigned? It must be a man with excellent reputation ,
otherwise everything will fail soon" . 19 The reply is not preserved but Olber s
refused .
Schumacher was right to demand certain qualifications from the future publisher of an astronomical journal . On the one hand he had to be a well-know n
astronomer trusted by his colleagues, on the other hand he had to be read y
to sacrifice much of his time for organisational and editorial work in th e
interest of the journal . And he had to be able to control this complicate d
project fully. In addition the publisher had to be able to evaluate criticall y
the manuscripts and to collaborate sensitively and consistently with the authors - he should have, according to Lindenau "along with thoroughnes s
and versatility of mathematical-astronomical education at the same time t o
a high degree the trust of the entire astronomical world" .2 0
Schumacher did not suspect that before long he would have to play thi s
important role . The first hint that Schumacher might be a suitable candidate can be found in a letter from Sept . 1819 from Lindenau to Schumacher
where referring to Zach's "Correspondance Astronomique" he writes : "D o
you really feel like taking over the publishing of such a journal in the Germa n
language?" . 21 But by the beginning of 1820 Lindenau thought that Olber s
was a desirable candidate, too . 22 A bit later they seem to have concentrate d
on Schumacher . Lindenau argued as follows : `"Since Schumacher combine s
a great literary erudition with diligence I might think that he is well fitte d
for such a job . i23 After the summer of 1820 many negotiations must hav e
been conducted behind the scenes among the astronomers as well as between Schumacher and the officials of the Danish King's government . As a
clear result in March 1821 Schumacher was officially requested to found a n
19 O1bers-Nachlaß, Staats- und Universitätsbibl . Bremen, Briefwechsel Schumacher Olbers, 01 1 ff ., 3 .9 .1819 ("Es muß ein Mann von erstem Rufe sein, sonst fällt alle s
bald zusammen ." )
20 Bessel-Nachlaß, Archiv der Berlin-Brandenb . Akademie der Wiss . Berlin, Nr . 287 ,
Lindenau, 1 .3 .1820 ("der neben der Gründlichkeit und Vielseitigkeit mathematisch astronomischer Bildung, zugleich auch das Vertrauen der ganzen astronomischen Wel t
in hohem Grade besitzt" )
21 Schumacher-Nachlaß, Lindenau .
22 Bessel-Nachlaß, Nr . 287, Lindenau, 1
.3 .1820 .
23 Lindenau an Gauß, 7 .7 .1821
. Niedersächs . Staats- u . Univ .-Bibl . Göttingen, Cod . Ms .
Gauß 101, Briefe A (Lindenau V), Brief Nr . 208 ("Da Schumacher mit einer grossen
litterarischen Bekanntschaft einen grossen Fleiss verbindet, so möchte ich glauben, das s
er zu einen solchen Geschäft gut geeigenschaftet sey" .)
MfM 46 :2
11 1
astronomical journal (see figure 3) . This request was made by Johann Sigismund von Mösting . Mösting was a highly educated person . Having studie d
jurisprudence in Copenhagen, he was the director of the Danish Reichsbank
in 1813 . Soon thereafter he became minister of finance, president of th e
Chamber of Finance and for many years the Prime Minister of the Danis h
Kings, chancellor of the Dannebrog-Order, in 1838 director of the King' s
Library and, last but not least, he was a friend and promoter of astronomy
as well . 24
About Mösting's intention to found an astronomical journal we know first
from Schumacher's information to Gauß on 1Vlarch 27, 1821 . Unfortunately,
the 1211 preserved letters from Mösting to Schumacher from 1821 til l
1845 started on June 12, 1821 . 25 The mentioned "request" by Mösting ,
probably in the form of a King's order, indicated to Schumacher tha t
he should "publish an astronomical newspaper in Altona with approximately one printed sheet per week serving the most lively communicatio n
among astronomers" . 26 In the preface to the first volume of Astronomisch e
Nachrichten dated Altona September 1821 Schumacher wrote : "By higher
support I am able to offer this journal to astronomers and mathematicians a s
a means of quick circulation of individual observations and short news" . 2 7
Schumacher had sent his colleagues a similar text already in June, 1821 ,
inviting them to collaborate .
At that time Schumacher was still formally a professor of astronomy i n
Copenhagen and he received his salary at least till 1845 . 28 But in connection with the foundation of the Astronomische Nachrichten he was allowed
to have Altona as his permanent residence and further to suspend his dutie s
at the university. This privilege must have been a basic precondition for him .
For the foundation and later the maintenance of the journal money was pai d
from the King's budget : "By publishing astronomical news and treatise s
our Minister of Finance wants to promote the rapid publication of scientifi c
works and therefore really covers all expenses", Schumacher wrote to Enck e
24 Jørgensen, Harald: Johan Sigismund v . Mösting . Dansk biografisk Leksikon, vol . 10 .
København 1982, 300-302
25 Schumacher-Nachlaß, Möstin g
26 Briefwechsel zwischen Gauss und Schumacher, vol . 1 . Altona 1860, 27 .3 .182 1
''Schumacher, H .C . : Vorwort . Astronomische Nachrichten 1 (1823), 1- 2
28 Københavns Universitet 1479-1979, vol . XII.1 . Det matematisk-naturvidenskabelig e
Fakultet . København 1983, 63
112
MfM 46 : 2
J. Hamel
ASTRONOMI.SCH E
II
A
,
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Neapel, London und Edinburgh, der astronota hen Gesellecb n ft in London, und der Kdnigliehe n
Lar,dbaushalludgagesclIsehall in Copcnbagen~ Lb nmdgLed der Soùetc oF useful arts in Edinburgh.
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Figure 3 : Astronomische Nachrichten, title page of volume 1
MfM 46 :2
Schumacher
mediator between Denmark and Germany
11 3
on January 26 1822 . 29 For the year 1849 there is the information that Schumacher received an annual financial aid of 640 Reichsbankthaler in order t o
cover the expense for the publication of the Astronomische Nachrichten. 30
During the first decades the publication of the Astronomische Nachrichte n
was possible only because of the generous support by Frederik VI and afte r
his death in 1839 by Christian VIII . It can be considered as an extraordinar y
example of international promotion of science . Looking back to history it is
difficult to find another time and place where it would have been possibl e
to publish permanently an astronomical journal . To underline this I woul d
like to refer to the circumstances under which the Astronomisches Jahrbuc h
had been published in the Prussian capital . In 1774 the Berlin Academy
founded an annual with a literary part because it was considered to b e
useful for the communication between astronomers . Very soon, however, th e
Academy no longer wanted to bear the financial risk and retreated from th e
project . Therefore Bode had to run it as a completely private enterprise .
It was extremely difficult to find a publisher for the supplement edition s
1
containing scientific articles because the financial profit was doubtful . 3
Johann Sigismund von Mösting (1759-1843) was the mediator in the projec t
between the Danish King and Schumacher . He was bound to Schumacher by
ties of deep friendship and gratitude . This is attested by the great number
of letters exchanged between them where they addressed each other wit h
the rarely used "Dear friend" . They often wrote about very private matters ,
but also about political affairs in Altona, revealing interesting insights int o
Schumacher's living conditions . On July 27, 1827, for instance, Schumache r
reported about an incident in Altona with a police officer von Aspern : "Fo r
the calm inhabitants of Altona this turmoil had the unpleasant effect tha t
the police have almost disappeared . Beggars rush into the houses and I
guess that thieves will not refrain from taking advantage of the situation .
I barricade myself every night, as well as I can with my chronometers an d
instruments in my house and I will try to defend the fortress ." 3 2
29 Encke- Nachlaß, Archiv der Berlin-Brandenb . Akademie der Wiss . Berlin, Schumacher ,
26 .1 .1822
30 Schumacher, H .C ., an J . Collin, 3 .3 .1849, quotation Nielsen, Axel V ., H .C . Schumacher, 29 6
31 Herrmann, Dieter B . : Die Entstehung, 23-2 4
32 Rigsarkivet . Privatarkiver J .S . Mösting, No . 6023, 27 .7 .1827 ("Für die ruhigen Einwohner Altonas hat dieser `tumult die unangenehmen Folgen, daß die Polizei fast ganz
darniederliegt . Die Bettler drängen sich jetzt in die Häuser, und ich vermuthe die Diebe
114
J.
Hamel
MfM 46 : 2
Mösting did not play the role of the willing financier but encouraged Schumacher to undertake the important and difficult publication of the journal .
Nevertheless, it seems to be a bit exaggerated when Schumacher on April 10 ,
1821 mentioned to Gauß that "everybody will believe that it is my idea, bu t
in fact it comes from our minister of finance and is not imposed on him" .3 3
In the first volume of the Astronomische Nachrichten Schumacher wrote a s
follows : "With the sincerest gratitude for the great measure of support fo r
my scientific plans I feel the rare happiness to live under a King like Frederik, loving and protecting in a King's manner the exact sciences, and unde r
a minister like Mösting, full of the same love, putting the great plans of hi s
King into practice ." 34 It was mainly Mösting who presented Schumacher' s
petitions and reports to the King and hurried to send the confirmation t o
Altona . Schumacher also had good relations with other officials of the cour t
such as Johan Gunder Adler (1784-1852), Royal Secretary of the cabine t
and secret state councillor . 3 5
But the relation to the court was kept mainly through Mösting . Schumache r
knew his ideas to be in good hands there . Most probably it is thanks t o
Mösting that a number of the most important astronomers working at Ger man observatories became Knights or even Commanders of the Dannebrog Order, as for instance Bessel, Olbers, Hansen and Gauß . It was Schumacher' s
unofficial duty to explain to his colleagues how to wear the "decorations "
and to write the letter of thanks to Mösting as the Chancellor of the Orde r
and the King . 36 Christian as well as Frederik allowed Schumacher to suggest who among his colleagues deserved to be decorated . 37 One may have
different opinions as far as decorations and medals are concerned, but th e
attitude of the Danish Court reflects the high esteem for scientific researc h
hardly to be found at any time in any other country.
In this connection the medal endowed by Frederik VI for the discovery o f
werden auch nicht unterlassen, auf die Conjunctur zu speculirn . Ich verschanze mich
jede Nacht, nach besten Kräften mit meinen Chronometern und Instrumenten in meine m
Hause und werde die Festung zu vertheidigen suchen ." )
33 Briefwechsel zwischen Gauss und Schumacher, vol . 1, 10 .4 .182 1
34 Schumacher, H .C . : Zusatz des Herausgebers
. Astronomische Nachrichten 1 (1823) ,
376
35 Schumacher-Nachlaß, J .G . Adler ; Rigsarkivet . Privatarkiver J
.G . Adler, Nr . 500 8
36 Briefwechsel zwischen Gauss und Schumacher, vol . 3, 382-383, 390-391 ; OlbersNachlaß, 01 17, 26 .3 .1825 ; the Dannebrog-Ridders-Ordens Statuter in : De Kongelig e
Danske Ridderordener og Medailler, vol . 1 . København 1950, 94-10 2
37 Privatarkiver J .G . Adler, 4.6 .1840
MfM 46 :2
11 5
comets by telescope should be mentioned, 38 (see figure 4) . The "Gold meda l
of 20 ducats" confirmed by Christian VIII in April of 1840 had a special
dedication : "The medal will be dedicated to the person who first discover s
a comet invisible by the naked eye but visible by telescope at the time o f
discovery and with unknown time of orbit" . 39 As had been the case wit h
the Astronomische Nachrichten, an international effect was intended . O n
the one hand the origin of the discoverer was of no importance, on the other
hand Frederik had formed an international jury consisting of Schumache r
and Francis Baily in London . For the case that the two could not agree a
referee was appointed . After the death of Olbers in 1840 Carl Friedrich Gauß
became the referee, chosen by Christian among Schumacher's proposals :
Bessel, Gauß and Herschel . 40 The medal was designed in 1834 by the Danis h
artist Christen Christensen41 under supervision of the important classical
sculptor Bertel (Alberto) Thorvaldsen .
The endeavours to intensively promote science in Copenhagen, of which th e
examples mentioned are only a few among many, were honoured with gratitude by astronomers . In 1823 Johann Franz Encke, successor of Lindenau at
the Seeberg-Observatory near Gotha and later outstanding director of th e
Berlin Academy Observatory, exclaimed : "Where else than in Denmark can
one nowadays find such a sense among the rulers to heavily support reall y
useful work?" 42 The astronomers expressed their gratitude to the King' s
minister of finance in a special way : In their classical works about the topography of the moon in 1837 Wilhelm Beer and Johann Heinrich 1Vlaedle r
named a lunar crater after him . 43 It is to be seen as "Mösting" A and B ; a
second object was named "Schumacher" . Moreover, Christian VIII in 184 0
was honoured as the first foreign ruler to be elected member of the Roya l
38 Schumacher, H .C . : Abbildung der Cometen-Medaille . Astronomische Nachrichten 1 1
(1834), 137f.
39 Schumacher, H .C . : Allerhöchste Bestätigung der Cometen Medaille . Astronomisch e
Nachrichten 17 (1840), 24 1
40 Briefwechsel zwischen Gauss und Schumacher, vol . 3, 36 5
41 Dansk Kunstnerleksikon, vol . 1 . København 1994, 502-503 ; Allgemeines Lexikon der
bildenden Künstler 6 (1912), 53 7
42 Encke, Johann Franz : Auszug aus einem Briefe . . . anden Herausgeber . Astronomische Nachrichten 1 (1823), 375
43 Beer, Wilhelm ; Mädler, Johann Heinrich : Der Mond nach seinen kosmischen und individuellen Verhältnissen . Berlin 1837, 306f., 204 ; Gondolatsch, Friedrich : Johan Sigismun d
von Mösting und der Mondkrater Mösting A . Die Sterne 22 (1942), 17-26
J. Hamel
116
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f
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. :_ . :L,
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MfM
46 : 2
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Figure 4a
Figure 4a,b : "Circulair" related the foundation of the medal for the discovery o f
comets by telescopes by Frederik VI ., 17 .12 .1831, with a postscript by Schumacher
to Encke (Encke- Nachlaß, Archiv der Berlin-Brandenb . Akademie der Wiss . Berlin ,
Schumacher, 6 .1 .1832)
MfM 46 :2
7/Li
4/4, . .m .
. 1
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Figure 4b
. .
11 7
118
J. Hamel
MfM 46 : 2
Astronomical Society in London . 44
Schumacher's Astronomische Nachrichten - the only international journa l
in the field of astronomy with its first 31 years edited by Schumacher himsel f
- made the observatory in Altona the centre of international relation s
between astronomers . The success was immense as immense as the amoun t
of work for the publisher! In a letter to George Biddel Airy from 184 5
Schumacher speaks of about 1500 letters per year he had to draft and writ e
himself. 45 This seems not to be an exaggeration . In the estate in the Berli n
State Library alone nearly 10 000 letters to Schumacher from more than 75 0
authors can be found . Shortly before the Astronomische Nachrichten were
started Bohnenberger mentioned to Schumacher : "I am extremely please d
that thanks to your astronomic news and articles astronomers will get int o
closer relations with each other again ." 46
These expectations were truly fulfilled . The Astronomische Nachrichte n
published with support of the Danish Kingdom, became a centre of international communication in the field of astronomy. All astronomers of
any importance published in this journal whether they came from Ger many, France, England, Russia, Italy, the Netherlands, the United States ,
Denmark, of course and so on . Thanks to his connections through correspondence Schumacher became an international "bureau of conference", a
scientific "news agency" . He received information and spread it via his As-
tronomische Nachrichten or by letters to his colleagues . In a letter to Jonas
Collin (1776-1861), deputy in the Ministry of Finance, Schumacher wrote :
"Altona now has become the centre of astronomic relations recognised eve n
by England and France" . 47 And the famous John Herschel, secretary of th e
Royal Astronomical Society in London, concluded in a letter to the Danis h
King of 1Vlarch 13, 1840 that Schumacher's journal, was "one of the mos t
remarkable and influential astronomical works, which have ever appeare d
and which, while operating more beneficially on the progress of its Scienc e
than any similar work of modern times", has "made Your Majesty's city o f
Altona . . . the astronomical centre of the civilised world" . 48 Nothing mor e
has to be said .
44Privatarkiver J
.G . Adler, 25 .2 .-4 .6 .1840, incl . the letter from J . Herschel to Christia n
VIII, 13 .3 .1840, and its German transl . by Schumacher
45 Nielsen, Axel V
., H .C . Schumacher, 274
46 Schumacher-Nachlaß, J .G .F . Bohnenberger, 2 .10 .182 1
47Nielsen, Axel V ., H .C .
Schumacher, 29 1
45 Privatarkiver J .G . Adler, 27 .5 .1840
MfM 46 :2
11 9
The Astronomische Nachrichten survived many difficult situations : Firs t
around 1848/50 the differences between Denmark and Germany concernin g
the status of Holstein ; 49 later the first and second World Wars, and finall y
the transfer of the centres of astronomical research from Middle Europe . In
1998 the journal appeared in its 319th volume . The editorial bureau is at
the Astrophysical Institute in Potsdam. 50
Appendix
Schumacher's application for a scholarship to visit foreign observatories, Alton a
1808 April 26, Rigsarkivet Kobenhavn . Fonden ad usus publicos 1810-1812 (fig .
2) .
An die Königliche Direction des Fonds ad usus publico s
Heinrich Christian Schumacher Doctor der Rechte bittet unterthänigst um Unterstützung in seinen astronomischen Studien durch Bewilligung des Reisestipendi i
zu einer astronomischen Reise .
Die Bitte mit der ich mich zu nahen wage scheint so kühn, daß ich ohne Gefah r
unbescheiden zu scheinen nicht die Entwickelung ihrer Motive übergehen darf, Mo tive die aus einer kurzen Darstellung meiner Lage hervorgehen .
Von Jugend an, durch Neigung zu den mathematischen Studien hingezogen, mußt e
ich nachher aus fremden Rücksichten, mich mit dem Rechte beschäftigen, aber nu r
um bald mit desto grösserem Eifer, mich zu den früheren Untersuchungen zu wen den . Ein überwältigtes Hinderniß beschleunigt, statt zu hemmen .
Kaum war es mir vergönnt aus der reinen Theorie in die Anwendungen zu treten ,
als vor allen die Astronomie mich anzog . Wenn andere Wissenschaften theils auf
dem, ietzigen Zustand der Ausbildung, theils sogar auf menschlichen Schwächen ,
und moralischer Unvollkommenheit gegründet, ihr ephemeres Daseyn mit diesen
Veränderlichen selbst wechseln, oder verlieren; so kann sie in der Natur gegründet ,
mit ihr von gleichem Umfange, gleicher Würde, nur mit dem Weltall veralten, un d
vergehen . Ihre leuchtende Bahn wird nur durch das Unendliche begränzt . Den n
wie die Naturwissenschaften in absteigender Linie, die Organisation bis zum Ato m
verfolgen, so geht sie in aufsteigender von der Erde zur Sonne, von der Sonne z u
ihrer Sterngruppe, von dieser zum Sternenheer .
49 Schwarz, Oliver ; Strumpf, Manfred: Peter Andreas Hansen und die astronomisch e
Gemeinschaft . Beiträge zur Astronomiegeschichte 1 (1998), 141-154 (Acta Historica Astronomiae ; 1 )
50 Present attempts to work scientifically with the Berlin estate of Schumacher so fa
r
have failed because of the lack of funding . Allow me to propose the idea to put this into
practice as a German-Danish joint project, in Schumacher's spirit .
120
J. Hamel
MfM 46 : 2
In der Ruhe des Landes, nach vollendeten academischen Jahren, beschäftigte ich
mich ausschließlich mit ihrem theoretischen Theile, zur Praxis führte nachher mic h
mein Freund der Professor Pfaff in Dorpat . Mit ihm beobachtete ich ein Jahr hin durch, das nur zu schnell verging .
Dann, ohne günstige Aussichten in Frankreich zu benutzen, fest überzeugt mei n
Glück sey an mein Vaterland gebunden, kehrte ich zurück voll Hoffnung nicht die
verlassenen juristischen Studien wieder ergreifen zu dürfen . Dennoch scheint mic h
iezt die Nothwendigkeit dahin zurückzuführen, da Mangel des Vermögens mich ver hindert eine künftige Anstellung in meinem Fache abzuwarten . Die einzige Hoffnung mit der ich iezt meinen Muth belebe, beruht auf eine gnädige Gewährun g
meiner Bitte um Unterstützung, bis sich eine für mich passende Anstellung ergebe n
sollte .
Um nun die Zeit bis dahin so nützlich wie möglich zu verwenden, wünschte ic h
die vornehmsten Sternwarten zu besuchen um alles kennen zu lernen, was iezt zu r
Vervollkommnung des beobachtenden Apparats geschehen ist, und zugleich mich mi t
den ersten Astronomen in nähere Verhältnisse zu setzen, Verhältnisse die mir be y
einer künftigen Anstellung von dem grössten Nutzen seyn würden .
Ich wage also meine Bitte mich durch gnädige Unterstützung in den Stand zu setzen ,
die ersten Sternwarten besuchen zu könne n
unterthänigst Altona den 26st . April 1808 . Heinrich Christian Schumacher .
MfM 46 :2
12 1
Viète's Generation of Triangle s
Otto B . Bekken *
Caspar Wessel's main aim in his On the Analytical Representation of Di rection is from given directed line segments to form others by algebrai c
operations and to form the product of two lines in the same plane as th e
positive unit :
As regards length, the length of the product should be the product o f
the length of the factors ,
and "the directional angle of the product, or its deviation from the
positive unit is the sum of the directional angles of the factors . "
As Branner and Voje Johansen have recently discovered (see [Wessel (1999) ,
44-49]), Wessel used complex numbers to represent line segments in his sur veying reports as early as 1787 - ten years before he wrote his theoretica l
memoir on the subject . Where he got his inspiration for this, we do not
know . His only reference on these matters reads "nobody else has treate d
it . An exception might be Master Gilbert in Halle, whose prize memoi r
on Calculus Situs may contain some explanation on this subject ." [Wesse l
(1999), 103] All attempts to find this prize memoir by Gilbert have so fa r
failed . l
I will here only recall some material from two papers by Francois Vièt e
(1540-1603) and his collaborator Alexander Anderson (1582-1619) . T .R .
Witmer's translation of 1983, which is our source, was based on Frans va n
Schooten's 1646 edition of Viète's manuscripts . In this context, these papers by Viète have been brought forward by Glushkov, by Bashmakova an d
Slavutin, by Bashmakova and by Itard (see the bibliography below) .
In Viète's Isagoge in Artem Analyticern we find a section entitled The Genesis of Triangles with his proposition XLVI and the accompanying figure s
* Kristiansand, Norway
' However, see Kirsti Andersen's argument on p .72 of [Wessel (1999)]
NIfM 46 : 2
BG + DF
IBG - DF I
Figure 1 : "To construct a third right triangle from two right triangles "
(see figure 1) . His reasoning, which is verbal, can be symbolized throug h
( ZH ) 2
= Z2H2 = (B 2 + D2 )(F2 + G2 )
=
B2 F2 + D2 G2 + B2 G2 + D2F2
B2F2 + 2BFDG + D2 G2 + B2 G2 + 2BGDF + D 2F2
(BF + DG) 2 + (BG + DF) 2
Only the sides of the new triangle are presented in his third triangle generated from two right triangles by either of these methods . Moreover, a right
triangle generated by the first method is called a synaeresic (taken together )
triangle, and by the second method a diaeresic (taken apart) triangle, "fo r
reasons set out in the proper place " , says Viète.
This proper place seems to be his Ad Angularium
Sectionum Analyticen
Theoremata which opens with two theorems
Theorem I
If there are three right triangles the
acute angle of the first of which differ s
from the acute angle of the second by
the acute angle of the third, the firs t
being the largest of these, the sides of
the third will have the likenesses of th e
diaeresic triangle made from the firs t
and the second .
Theorem I I
If there are three right triangles and th e
acute angle of the first of these plu s
the acute angle of the second is equa l
to the acute angle of the third, the side s
of the third will have the likenesses o f
the synaeresic triangle made from th e
first and the second .
Viète 's Generation of Triangles
MfM 46 :2
12 3
The acute angle is the angle between the hypotenuse and the base, an d
the figure above has been amended by adding symbols to these angles . The
demonstrations given by Anderson are based on Euclidean arguments, bu t
can easily be interpreted as proofs of the trigonometric formulas for the sine
and the cosine of the difference and the sum of two angles .
Viète's theorems say that if we want the acute angle of the product to b e
the sum of the acute angles of the factors, then we must use the synaeresi c
product of triangles wit h
base
BF - DG,
and perpendicular
BG + DF,
and if we want the acute angle of the product to be the difference of the acut e
angles of the factors, then we must use the diaeresic product of triangle s
with
base
BF + DG,
and perpendicular
BG - DF.
The reader skilled in complex number multiplication, can see the analog y
to the real and imaginary parts of the product of the complex number s
B + D-V-1 and F + G A/-1 in the synaeresic case, and the product o f
B + D,/-1
with the complex conjugate F - G-/-1 in the diaeresic case .
The similarity between Viète's product of triangles and Wessel's product o f
directed line segments is also quite apparent, but whether Wessel had acces s
to or had seen these papers by
Viète, this we do not know .
Reference s
Bashmakova, Isabella G . : "Diophantine Equations and the Evolution of Algebra", Trans lation of Proc . ICM Berkeley 1986, AMS Transi . (2) Vol . 147 (1990), 85-100 .
Bachmakova, Isabella G . & E .I . Slavutin : "Genesis Triangulorum de Francois Viète et ses
recherches dans l'analyse indéterminée", Archives for History of Exact Sciences 1 6
(1977), 289-306 .
Branner, Bodil & Nils Voje Johansen : "Caspar Wessel (1745-1818) . Surveyor and Mathematician", in Caspar Wessel On the Analytical Representation of Direction, Copenhagen : Royal Danish Academy of Sciences and Letters, Matematisk-fysiske Meddelelser 46 :1 (1999) .
Glushkov, Stanislav : "An interpretation of Viete's 'calculus of triangles' as a precursor o f
the algebra of complex numbers", Historia Mathematica 4 (1977), 127-136 .
Itard, Jean : "Matériaux pour l'histoire des nombres complexes", La Bibliothèque d'Information sur l'Enseignement Mathématique 1968 .
124
O.B. Bekken
MfM 46 : 2
Viète, Francois : The Analytic Art, trans . T . Richard Witmer, Ohio: Kent State University
Press 1983 .
Wessel, Caspar : On the Analytical Representation of Direction, Copenhagen : Royal Danis h
Academy of Sciences and Letters, Matematisk-fysiske Meddelelser 46 :1 (1999) .
MfM 46 :2
12 5
Argand and the Early Work on Graphical
Representation: I\ ew Sources and Interpretation s
Gert Schubring *
The aim of the present paper is to investigate the works of some of th e
mathematicians who dealt with the graphical representation of the comple x
numbers independently of Wessel and in particular with the most famou s
of them : Argand . However let me begin by mentioning some of the basi c
concepts in Wessel's paper in order to establish connections with the ongoin g
discussion in the mathematical community at large and more specifically i n
France .
Wessel was not concerned with epistemological problems associated with th e
idea of negative quantities in the French discussion - rather, he extracte d
from it two basic notions and rearranged them : the notion of être numériqu e
and the notion of être spécifique or qualité of a given quantity. Whereas thes e
two notions had been treated as separate concepts in the French discussion s
of the 18th century, Wessel forged them together renaming them as lin e
segments and directions of lines .
His innovative procedure was to enlarge the notion of direction . He con sidered the concept of simply opposed directions like positive and negative
directions as well-known and envisaged going beyond the restriction to di rections along the same line, extending the concept to directions in the entir e
plane and even in the sphere . As Wessel explains :
if one takes the operations in a wider sense, and does not, as before ,
restrict them to be used only on segments of the same or opposit e
direction, but extends their formerly restricted concept somewhat, s o
that it becomes applicable not only in the former cases, but also i n
infinitely many more cases (Wessel 1999, 95) .
*Institut für Didaktik der Mathematik, Universität Bielefeld, Postfach 100131, D 33501 Bielefeld, Germany
G. Schubring
126
MfM 46 : 2
An example of a generalization proposed by Wessel based on this approac h
is the perpendicular unit.
My intention is not to analyse the application of these generalizations but t o
discuss the point raised by Valentiner, one of the editors of the French version, in 1897 . Valentiner expressed his astonishment about the innovation s
achieved by mathematical "outsiders" in these words :
il est étonnant qu'un homme puisse composer un ouvrage aussi remarquable que celui qui nous occupe, après avoir dépassé la cinquantaine ,
sans avoir jamais, ni avant ni après, produit aucune oeuvre scientifique
(Valentiner 1897, V) .
Almost the same remark can be made about most of the inventors of th e
graphical representation of complex numbers before Gauß's publication o f
1831 . In fact, historians of mathematics frequently wonder about the simultaneity of these inventors around 1800 and about their common pattern o f
marginality to the mathematical communities, and of amateurism .
My intention is to contribute somewhat to an understanding of this mystery. Two authors who provide some elements for illuminating the issue ar e
Argand and Buée .
1
Argand
Argand, the author of the booklet Essai sur une manière de représenter le s
quantités imaginaires dans les constructions géométriques, is usually identified as Jean Robert Argand, who lived from 1768 to 1822 and is reporte d
to have been a bookkeeper in Paris (teneur de livres) . As we will see, thes e
few known data seem to be doubtful .
Argand's text, allegedly published in 1806, did not remain unknown lik e
Wessel's, but was discussed beginning in 1813/14, with Argand himsel f
participating in the discussions . These debates took place publicly in th e
Annales de mathématiques pures et appliquées, the first journal which specialised in mathematics, published by J . D . Gergonne at Montpellier . Yet ,
the public of this journal was a restricted one, as the dominant Parisia n
mathematicians did not participate in it . l Like Wessel, Argand begins wit h
' Lacroix, for example, did not communicate the fact that Buée's paper had bee n
published in 1806 to Gergonne and his journal but rather to a colleague of Gergonne's .
Lacroix's note became known to the Annales (vol . 4, 1814, 367) only by this intermediary .
MfM 46 :2
Argand on Graphical Representation
12 7
reflections on oppositely directed quantities and uses the analogous conceptual differentiation of rapport numérique and rapport de direction . He goe s
on to ask if one can generalize these concepts to imaginary quantities, give n
that one cannot assign them a geometrical representation on one line as i s
the case for two opposed, so-called positive and negative directions . Thus ,
he looks for a geometrical construction in the same plane . He proposes such
a construction in the plane by interpreting the proportio n
1 :-1=-1 : 1
as yielding the mean proportional . By this approach, Argand transforme d
the proportion 1 : -1 = -1 : 1, which up until his time had been the touch stone for every approach to understand negative numbers, into the corner stone of a new theory. Argand is therefore a nice illustration of Lakatos' s
model of conceptual development : former monsters become the definitiona l
foundation for innovations . Having identified the notions of grandeur ab-
solue and of direction as constituting his conceptual basis, he investigate s
whether it would be possible to combine them so as to assign a place to th e
imaginary quantities within the conceptual field of positive and negativ e
quantities ("une place dans l'echelle des quantités positives ou négative s
(Argand 1874, 6) .) His answer was to exploit the basic proportion in a geometric way :
En y réfléchissant, il a paru qu'on parviendrait à ce but si l'on pouvai t
trouver un genre de grandeurs auquel pût s'allier l'idée de direction ,
de manière que, étant adoptée deux directions opposées, l'une pour le s
valeurs positives, l'autre pour les valeurs négatives, il en existât un e
troisième telle, que la direction positive fût à celle dont il s'agit comm e
celle-ci est à la direction négative (ibid . )
In expanding on this approach, Argand did not restrict himself to perpendicular lines, to imaginary quantities . Hence, he instead constructed al l
directions in a given plane geometrically and thus achieved the constructio n
of complex numbers as well . Argand gave as the general algebraic form fo r
complex numbers
+a+ b1/-1 (ibid ., 12 )
and showed, using figure 1, the corresponding lines with their direction .
Leading mathematicians seemed - during this period - not to be very
128
G . Schubring
MfM 46 :2 .
Figure 1 : Argand 1874, 7
interested in these foundational efforts . A revealing example is presented b y
Adrien Marie Legendre (1752-1833) . As is well known, Argand's ideas
attracted some public interest for the first time when Jacques Frédéri c
Français (1775-1833), an Alsatian mathematician, published an article i n
Gergonne's Annales in 1813 where he referred to the ideas of an unknow n
author :
Je dois [ . . . ] à la justice de déclarer que le fond de ces idées nouvelle s
ne m'appartient pas . Je l'ai trouvé dans une lettre de M . Legendre à
feu mon frère [François Joseph Français, 1768-1810], dans laquelle ce
grand géomètre lui fait part (comme d'une chose qui lui a été communiquée, et comme objet de pure curiosité) du fond de mes définitions
2 e et 3 e , de mon théorème IeT , du corollaire 3e de mon théorème II e
[
. ] . Je désire que la publicité que je donne aux résultats auxquel s
je suis parvenu puisse déterminer le premier auteur de ces idées à se
faire connaitre, et à mettre au jour le travail qu'il a fait lui-même su r
ce sujet (Français 1813a, 71) .
As is likewise well known, this appeal induced Argand to enter into th e
debate and to show that he was the author .
I have been lucky enough to find this letter from Legendre to Français th e
elder brother . It is a letter dated 2 November 1806 ; in its final part, Legendr e
reports to Français that an unknown person had given a mémoire to him .
Legendre was astonished by the quality of this paper and by the lack o f
ambition of its author .
MfM 46 :2
12 9
Il y a des gens qui cultivent les sciences avec assez de succès san s
être connus et sans courir pour la renommée . Dernièrement j ' ai v u
un jeune homme qui m'a engagé à lire un travail qu'il avait fait sur
les imaginaires ; il ne m'expliquait pas très bien son objet, mais il m e
faisait entendre qu'il regardoit les quantités dites imaginaires comme
aussi réelles que les autres, et qu'il les representait par des lignes . J'ai
témoigné d'abord bien des doutes à l'auteur, cependant j'ai promi s
de lire son mémoire . J'y ai trouvé contre mon attente, des idées asse z
originales, fort bien présentées, appuyées de connaissances de calcu l
assez profondes, et enfin qui conduisent à des conséquences fort exacte s
telles que la plupart des formules de trigonométrie, le théorème d e
Cote, etc . Voici une esquisse de ce travail qui vous intéressera peutêtre et qui vous fera juger du reste . 2
Legendre then explained Argand's approach for the geometric constructio n
of imaginary and complex quantities and went on to outline two of Argand ' s
applications :
trigonometric formulas such as
cos(a + b) = cos a cos b - sin a sin b ,
- Cotes's theorem (also called de Moivre's formula) :
cos(na) + \/-1 sin(na) _ (cos a + \/-1 sina) ' .
Legendre declared himself not to be interested in this as a research subjec t
but appealed to Français to further develop these ideas :
Je ne rends ici qu'une petite partie de ses idées, mais vous y suppléerez
et peut-être vous trouverez comme moi qu'elles sont assez singulière s
pour mériter attention . Au reste je vous les abandonne simplemen t
comme objet de curiosité et je ne me chargerois pas de les défendre .
It seems that without this letter from Legendre to Français, Argand's pape r
would neither have been discussed by contemporary mathematicians no r
would it ever have become publicly known . In fact, there is no proof that
this paper was published in 1806 . The only indication of this date is give n
by its appearance on the title page of the printed book . This is not decisive ,
for three reasons :
2 In my talk, I showed a copy of this letter, which belongs to a private collection.
130
G. Schubring
1VIfM 46 :2 .
The booklet was not deposited at the Dépot Legal at Paris as pre scribed by the law for published books, as my research at this offic e
has shown .
In all my searches I was unable to identify an original version in a
French or a foreign library . All the copies in libraries are either th e
1874 second version published by Hoüel or later reprints . And th e
1874 edition is made on the basis of a copy which Argand had sent t o
Gergonne .
As I have learned recently from Luigi Pepe (Ferrara), there is, how ever, at least a second copy of the first printing . It is in the possessio n
of Jean-Luc Verley (Paris), who, as a bibliophile mathematician, possesses an impressive collection of old books . Curiously enough, a n
inspection of this copy proved that it belonged to Gergonne, too! 3
Legendre ' s letter gives a supplementary argument : Legendre would n ' t
have communicated Argand's ideas to Français and invited him t o
pursue and further develop them if the paper had already been published or if publication were imminent . Moreover, according to the
rules of the Paris Institut, only unpublished manuscripts were allowe d
to be fully reported upon in this Academy . Thus if Argand wanted t o
have a chance of such a report, his paper had to remain a manuscript .
In fact, Argand himself mentioned in 1813 that Legendre examined ,
in 1806, "mon manuscrit" (Argand 1813b, 133) .
The only thing which one can say with certainty is that the paper wa s
printed somewhere between 1806 and 1813, without an indication of th e
author ; it was, however, only privately distributed 4 and not published .
Even in 1813, it did not attain the character of a real publication, as th e
only known copies all belonged to Gergonne (see above) . Although Argand
had offered, in his 1813 paper in the Annales, to sell copies of his Essai
(Argand 1813b, 133), 5 apparently no other contemporary mathematicia n
3I am very grateful to Jean-Luc Verley for his cooperation and assistance in this re search . There is a manuscript note in his book which states that this copy had belonge d
to Gergonne and that it was used for Hoüels reprint . Since Verley's copy does not bear
Argand's dedication to Gergonne of 1813 (written on the page with the printer's data, opposite to the title page), which is reproduced in the reprint, it is clear that this dedicatio n
was taken by Hoüel from a second copy originally belonging to Gergonne .
4 As Argand affirmed himself in 1813, he had distributed the printed version only i n
a "très-petit nombre" (Argand 1813b, 133), probably to some friends . In this passage h e
characterized his paper as only being an "écrit" (ibid .) .
5 On the title page of the copy which he dedicated to Gergonne, he had crossed out the
MfM 46 :2
13 1
contacted Argand or ordered the booklet . Even Français, most interested
in the text, did not address Argand and rather preferred to borrow a cop y
from Gergonne (cf . Français 1814b, 222) . It was only due to Hankel's reviva l
of interest in Argand's original paper (Hankel 1867, 82) that Hoüel was
motivated to search for it and to reprint it .
Besides the extraordinary history of Argand's
mémoire, his biography of fers at least as many questions . In fact, Legendre's letter has caused me t o
question all the biographical information available on Argand, which is actually quite scarce . I began to wonder about the fact that Legendre calls th e
paper's author "un jeune homme" . Legendre was 54 years old in 1806 an d
the Argand of the standard sources already 38 years old . Can one believe
that Legendre would, at age of 54 years, call a man of 38 years a "youn g
man"? This is at least doubtful, the more so since Legendre's first sentenc e
seems to imply that he rated Argand as capable of a scientific career if h e
had been interested in it .
Almost all the available biographical information on Argand is due to Hoüel
who undertook some research on the matter for his 1874 edition . All this information is based on Hoüel's assumption that Argand originated in Genev a
(Hoüel 1874, ix) . Hoiiel gave no justification for this claim, and one can onl y
guess that he came to this assumption by induction : Ami Argand (17501803), an inventor active in physics and chemistry, known for the construc tion of a lamp, who had lived a certain time at Paris, was in fact born i n
Geneva . This hypothesis led Hoüel to address colleagues at Geneva and t o
ask them to conduct biographical research . They came up with the information on birth date, first names and profession of a certain Jean-Rober t
Argand (ibid ., xv-xvi) . Although they added a cautionary remark about th e
identity ("C'est très probablement l'auteur du Mémoire de Mathématique s
en question", ibid .), their information has been taken up to now as pur e
truth, in all related historical publications and in all biographical dictionar ies . 6 The article on Argand in the Dictionary of Scientific Biography, fo r
distributor's address and indicated his own so that one should order copies of the bookle t
there .
Only once, a doubt has been raised by an anonymous person (again!) in the journa l
L'intermédiaire des Chercheurs et Curieux : in 1875, one "T . de L ." remarked that one
has the biographical data of this Jean-Robert Argand, "mais on n'a pas la certitude qu'i l
s'agisse lå de l'auteur de l'Essai sur une marnière de représenter• les quantités imaginaire s
[ . . . ]" (8e année, 1875, 424) . He received two answers : the first declared Ami/Aimé
Argand to be the Essai's author (9e année, 1876, 688) and the second only repeated
132
G. Schubring
MfM 46 : 2
instance, shows no hesitations or doubts and even reports that the "verification of the dates of his birth and death is given by H . Fehr in the
Intermédiaire des mathématiciens" in 1902 . An inspection of this sourc e
shows, however, that Fehr had started from the conviction of correctness o f
Hoüel's hypothesis and just checked in the "Archives de l'État de Genève "
the death date of that Jean-Robert Argand born in Geneva . ?
What objective information about Argand do we have? Unfortunately, w e
do not even have any primary evidence as to Argand's first name : his letters
as printed by Gergonne show only his last name . Also the tables of content s
of Gergonne's Annales cite him by his last name only. The only objective
information is his address in 1813 in Paris . Unfortunately again, as a resul t
of the fights of the communards in Paris 1870/71, most of the registers of
the administration of Paris are lost and it is therefore impossible to infe r
anything about the citizens living there from an address alone . The deat h
registers were also burnt . 8 Since historians of the nineteenth century faile d
to do adequate biographical research, it will now be quite complicated t o
obtain better information . One can, at least, infer from Legendre's lette r
that in 1806 Argand was unknown in the scientific community . Moreover ,
it is clear from his immediate reactions in 1813 and 1814 that he had eas y
and regular access to scientific journals . Apart from this we can only stat e
in terms recalling medieval history
that Argand "flourished" in 1806 ,
.
1813, and 1814
As regards his profession, there are two clues which point in the same di rection . Firstly, the address for distribution as given in the original printing is the shop of a clockmaker . And secondly, the very first paper which
Hoüel's affirmations (ibid ., 715) . These sources are mentioned in the (relatively) new:
Dictionnaire de Biographie Française, t. III, 1939, 504 .
7 L'Intermédiaire des mathématiciens, 9(1902), 74 . Fehr had answered to a query by G .
Eneström in 1900 in this journal ; Eneström had remarked that the death date of Argan d
was unknown . This unique journal functioned in an analogous way to the electronic mail newsgroups of today . The journal "L'intermédiaire des chercheurs et curieux" was
a somewhat parallel journal addressing all the sciences and the arts . As 1 have learnt
from the Archive d'État of Geneva, Fehr had more information at his disposal than h e
published later in his answer to L'Intermédiaire : the Archive had communicated to hi m
in 1901 - quoting from the acte de décès submitted to the Geneva registration office i n
1822 by the mairie of the tenth Paris arrondissement
that Jean-Robert Argand had
been an employee at the hospital of the Garde Royale . Had Fehr been afraid that thi s
additional information would falsify this person's authorship ?
8 Communication by the Direction des services d'Archives de Paris .
MfM 46 :2
13 3
Argand had sent for publication - his 1813a, written in February 1813 ,
occasioned by a problem announced in the Annales in 1812, and publishe d
several months before Français's paper (1813a) which opened the debat e
about Argand's Essai - discusses technical constructions of artisans fro m
an expressly theoretical point of view. Argand shows himself in this paper as
closely familiar with all the details of technical production, in particular o f
portable thermometers in the form of clocks, and as knowledgeable in recen t
scientific publications . For example, he quotes Laplace's Système du Mond e
in order to criticize certain shortcomings of practitioners (Argand 1813a ,
39) . In his conclusion, Argand pleads against the trial-and-error practice s
of artisans, which devalue the work of even the highly skilful, and plead s
for guidance by theoretical principles . His claim to have achieved better
practical results by such guidance (ibid ., 41) suggests that Argand was a
scientifically oriented technician, based in the Parisian clock industry .
Figure 2 : Figure of Argand's construction of a portable thermomenter (Argan d
1813a, Planche I)
It is useful to try to create a reconstruction of the events around Argand' s
Essai. A fairly probable one is the following . In the autumn of 1806, Legendre was approached by Argand, who tried to outline the main results in hi s
manuscript to him in a direct conversation . Legendre responded with skepticism as to the method and its applications . Upon leaving, Argand urged
Legendre to read his manuscript . Legendre had not retained the name of this
134
G. Schubring
MfM 46 : 2
man and assumed that the manuscript would show the name of its author .
When Argand had left, Legendre realized that the paper indicated neithe r
the address nor the name of the author . Upon reading the Essai, Legendr e
noticed its quality, he waited for a further visit from its author, but the author did not appear again . In order to end his own involvement with thes e
conceptions he wrote the report to Français in the letter of 2 Novembe r
1806 . Since Legendre firmly asked not to be bothered with discussions o n
this paper, neither the elder nor later the younger Français dared to ask hi m
about the paper or its author . On the other hand, Argand - apparently a
shy man - abstained from publishing his paper, due to Legendre's uninterested and sceptical reaction . Only the quite indirect reception of his idea s
via the brothers Français induced Argand to organize a later printing wher e
he arranged for the date of its composition to be put on the title page .
2
Adrien-Quentin Buée
The third author productive around the same period is Adrien-Quenti n
Buée (1748-1826), a French Catholic Priest a "prétre réfractaire " as th e
French during the Revolution called those priests who refused to give thei r
oath on the Constitution . Buée fled in 1792 to England and returned to
France in 1813 . His paper Mémoire sur les quantités imaginaires was read
in 1805 at the Royal Society and was published in 1806 .
Buée's paper is remarkable in its systematic evaluation of the earlier Frenc h
discussion on the nature of negative and imaginary quantities . He achieves
the establishment of a conceptual connection between the two basic concept s
of length or absolute value and of direction which had been separated s o
systematically in France during the 18th century .
Buée's achievements are likewise important for the conceptual developmen t
of the negative numbers and for the graphical representation of the comple x
numbers .
The important step in Buée's approach is that he clearly distinguishes between the two different meanings of the signs plus and minus, namely a s
- signs of operations, on the one hand, an d
signs of qualities of the quantities themselves, on the other . As he
notes :
MIM 46 :2
13 5
Des Signes "+" et "- "
Ces signes ont des significations opposées .
Considérés comme signes d'opérations arithmétiques, "+" et "- "
sont les signes, l'un de l ' addition, l'autre de la soustraction .
Considérés comme signes d'opérations géométriques, ils indiquen t
des directions opposées (Buée 1806, 23) .
Buée interpreted this sign of quality in geometrical terms, as direction, while
he attributed length to an arithmetical meaning . His important step was
not restricted, however, to this conceptual clarity of distinguishing betwee n
what he called arithmetical operation and geometrical operation, rather his
decisive step was to propose to unite both operations :
Lors donc qu'on réunit ces deux opérations, on fait réellement une
opération arithmético- géométrique ,
Buée immediately applied his approach to the investigation of imaginar y
quantities and hence expanded the applicability of the novel arithmeticogeometrical operations :
Je mets en titre, Du signe
et non De la quantité ou De l'unit é
imaginaire \/-1 ; parceque ,/-1 est un signe particulier joint à l'unité
réelle 1, et non une quantité particulière . C'est un nouvel adjectif joint
au substantif ordinaire 1, et non un nouveau substantif (ibid ., 27) .
, est le signe de la PERPENDICULARITÉ, dont la proAinsi
priété caractéristique est, que tous les points de la perpendiculaire
sont également éloignés de points placés à égales distances, de part
et d'autre de son pié . Le signe -/-1, exprime tout cela, et il est le seul
qui l'exprime (ibid ., 28) .
This novel approach where the perpendicular direction is treated in a manner analogous to the two traditional opposed directions on a line, is wel l
explained by Buée's following figure (see Figure 3) .
Like Wessel and Argand, Buée was no professional mathematician . As a
Catholic priest exiled in Britain, he engaged in publishing political-religiou s
tracts . Except for his paper on the imaginaries, his sole other scientific pape r
is on physics . And as Valentiner did in the case of Wessel, one can wonde r
how an author of political-religious tracts could arrive at such a level o f
clarté regarding the foundations of mathematics .
In order to further complicate the matter one can add that there was even a
fourth inventor, active in about the same period : Henri-Dominique Truel .
136
G . Schubring
MfM 46 : 2
Figure 3 : Figure 1 (Buée 1806, planche annexée )
Truel's existence and activity are asserted by Cauchy, who attributed the
priority of the discovery of the graphical representation to him . Truel, who
is not known otherwise, was called "un savant modeste" by Cauchy, who
reported that Truel had found the greatest part of Buée's and Argand' s
results by 1786 but that he had concealed them in his manuscripts . B y
1810, Truel had communicated his research to a marine engineer, Augusti n
Normand, at Le Havre, and it seems that Cauchy learned via Norman d
during his engineer's service at Cherbourg
of Truel's work and of thi s
graphical representation in particular (Cauchy 1847/1938, 175) .
3 François Daviet de Foncenex
Fortunately enough, Buée has given us a clue as regards the source of hi s
innovation . I have to confess that I noticed this hint quite late since this i s
a remark at the end of his paper ; maybe other authors missed it entirely
since it is mentioned nowhere in the literature . This indication is given as a
"Postscriptum" :
Since I wrote this mémoire, I have read - in the first volume of th e
Turin Academy Transactions a paper by M . Foncenex with the title :
Quantités Imaginaires where one finds the following
paragraph (Buée 1806, 83 ; my translation, G . S .) .
Refléxions sur les
He quoted this paragraph 6 completely . Due to its importance for our investigation, I will quote it completely, too :
MfM 46 :2
13 7
6 . Si l'on réfiechit sur la nature des racines imaginaires, qui comme on sait impliquent contradiction entre les données, on concevra évidemment qu'elles ne doivent point avoir de constructio n
Géométrique possible, puisqu ' il n'est point de maniére de les con sidérer, qui lève la contradiction qui se trouve entres les données immuables par elles mêmes .
Cependant pour conserver une certaine analogie avec les quantités
négatives, un Auteur dont nous avons un cours d'algébre d'ailleurs
fort estimable a prétendu les avoir prendre sur une ligne perpendiculaire à celle où l'on les avoit supposé, si par exemple (pl . I . Fig . I) on
devoit couper la ligne AB = 2a de façon que le rectangle des partie s
x x (2a-x), fut égal à quantité 2a 2 on trouveroit x = a+ \/(-a2 ), pour
trouver donc cette valeur de x, qu'on prenne sur la ligne AB, la partie
AC = a partie réelle de la valeur de x, & sur la perpendiculaire ED le s
CE, CD aussi = a, on aura les points D, E qui resolvent le problème
en ce que AD x DB, ou AE x EB = 2a 2 , mais puisque les points E ,
& D sont pris hors de la ligne AB, & qu'une infinité d'autres point s
pris de même, auroient aussi une propriété semblable, il est visible ,
que si cette construction ne nous induit pas en erreur, elle ne nous fai t
absolument rien connoître, c'est cependant là un des cas ou elle pourroit paroître plus spécieuse, car le plus souvent on ne voit absolument
pas comment le point trouvé pourroit résoudre la question, quelque s
changemens qu'on se permit dans l'énoncé du problême .
Les racines imaginaires n'admettent donc pas une construction géométrique, & on ne peut en tirer aucun avantage dans la résolution des
problèmes : on devroit par conséquent s'attacher à les écarter autan t
qu'il est possible des équations finales, puisque prises dans quel sen s
que ce soit, elles ne peuvent pas résoudre la question, comme les racine s
négatives, dont toute la contradiction consiste dans leur maniére d'être
à l' égard des positives (Foncenex, 1759, 122-123) .
Who was this Foncenex? François Daviet de Foncenex (1733/4-1799) wa s
an officer, living at Turin, interested in the sciences and in engineering .
He has published - besides the paper on imaginarres several papers o n
physics and technology . He is said to have been a friend of Lagrange durin g
Lagrange ' s stay at Turin and one even reports that Foncenex ' 1759 paper
expressed thoughts of Lagrange .
This sixth paragraph leads to at least three questions :
G. Schubring
138
MfM 46 : 2
an analysis of its mathematical meaning ,
the reception of this part of the paper ,
the identification of the algebra textbook mentioned at the beginnin g
of the 6th section .
As regards the analysis of the text, one should note beforehand that it i s
not mentioned in the historical literature not even by Cajori who is the
only one to extensively discuss other parts of the
mémoire. 9
Looking concretely at the text, it must be said that it is ambiguous an d
even contradictory . This character becomes visible only by considering th e
figure which is printed in another part of the volume :
E
Figure 4 : Foncenex 1759, fig . 1 : planche annexé e
On the one hand, Foncenex clearly shows how one can construct the imaginary quantities geometrically, and visualizes this construction by means o f
this figure . In fact, transforming the equatio n
x x (2a - x) = 2a 2
for x immediately yields :
x=a+
a2 -2a2 =a+ia.
In the text, Foncenex assures us that the construction does not lead t o
errors .
9 The only exception is Pieper, who just mentions Foncenex's interdiction verdic t
(Pieper 1984, 214) .
MfM 46 :2
13 9
On the other hand there is the commenting text before and after the constructive part . This commentary sets up signs saying that such numbers ar e
"off limits!" For instance, in the introductory comment, Foncenex gives an
epistemological argument :
Evidently, one understands that imaginary roots do not admit a possible geometric construction
and in the concluding comment :
imaginary roots do not, hence, admit a geometrical construction, an d
one cannot infer any advantage in resolving problems by using them :
one has therefore to pay careful attention to eliminate them, as fas as
it is possible, from the final equations (my emphasis, G . S .) .
The mathematical legitimation which Foncenex gave for his refutation i s
rather weak : He concedes that the construction is correct but argues that
one learns nothing new from it . And he adds that one can construct an
infinity of other points with analogous properties .
This analysis of the sixth section already leads to an at least partial answe r
to the second question : Among the readers of Foncenex's mémoire, there
certainly were some who became more convinced by the force of the visualization of a geometrical construction and who did not let themselves b e
deterred by epistemological interdictions . It is quite probable that one might
find, among such readers, marginal figures who as autodidacts might hav e
been less permeated by the dominant epistemology than those who adhere d
professionally to the mathematical norms .
mémoire were numerous : Foncenex had written this paper to investigate the difference of opinion between d'Alembert and Euler about the admissibility and the meanin g
And one has to know that the readers of Foncenex's
of logarithms of negative quantities (cf. Youschkewitsch, Taton in : Eule r
1980) . 10 I cannot enter here into a presentation of this year-long controversy which reveals highly illuminating epistemological dimensions of basi c
mathematical concepts . However I should mention that they ended this con troversy by exhaustion, without either of them convincing the other . Th e
fact that two famous mathematicians had not been able to reach a consensu s
'°The strong reception of Foncenex's paper is due to d'Alembert's paper of 1761 where
he made his controversy with Euler publicly known for the first time . In an appendix to this
paper written already in 1752, d'Alembert extensively discussed Foncenex's arguments .
140
G . Schubring
MfM 46 : 2
about basic concepts, attracted the attention of a great many people durin g
the second half of the eighteenth and at the beginning of the nineteent h
century who tried to develop their own solution . The problem seemed t o
be accessible by "common sense" and attracted, hence, in particular th e
interest of amateurs . Almost everyone who worked on the problem and subsequently wrote about it stumbled on Foncenex'
mémoire and read it . 1 1
I can now pass on to the third question : who is the author not named
by Foncenex who published a highly esteemed algebra textbook where th e
construction of imaginary roots on a perpendicular line was proposed ?
Foncenex's paper led me to assume that he had an author of his own time i n
mind . Consequently, I consulted a great many algebra textbooks of his tim e
and of the entire first half of the eighteenth century ; I consulted even genera l
textbook series for mathematics since they contain parts on algebra, too . I
detected, however, no author who had discussed or allowed such geometrica l
constructions . My subsequent evaluation of Robin Rider's bibliography o f
publications on algebra gave no better result (cf. Rider 1982) .
4 John Wallis
The only solution seems to be that Foncenex alluded to John Wallis's algebr a
treatise, published in 1685 in English and in 1693 in Latin, the latter in a
somewhat extended form . This first proposal for a geometric construction
is relatively well known so that I will just recall the structurally importan t
points .
On the one hand, Wallis denied there that quantities less than zero can exist .
On the other hand, he added that such a supposition is nevertheless no t
useless and not absurd ; in fact, Wallis continued to freely develop operation s
with negative quantities (Wallis 1693, 286) .
Using this conceptual basis, Wallis was the first to admit a geometrica l
construction of imaginary roots by interpreting them as mean proportionals :
" Buée, e .g .,
protested against Foncenex's principle of excluding imaginary roots (Bué e
1806, 85) . Buée's claim to have read Foncenex's paper only after completing his own on e
seems to be a " Schutzbehauptung" : In quoting Foncenex's paragraph no . 6, he refers to his
own - quite analogous - figure ; Buée has not only renamed the points within that quot e
according to his own figure but also tacitly corrected a misprint in Foncenex's origina l
text .
MfM 46 :2
14 1
Firstly, he explained the well known fact that i/bc is the mean proportiona l
between +b and +c , and then he formally extended this interpretatio n
to ANC as mean proportional between -b and -c . Eventually, he rather
postulated than demonstrated that i/- be can also have the signification of
the mean proportional between +b and -c . Likewise, he interpreted i/- b e
as the mean proportional between -b and +c (ibid ., 287) .
Wallis exemplified his algebraic argumentation by several geometrical constructions . As an introduction to his examples, he even seems to have bee n
the first who visualized the quadrants of the plane with their respectiv e
signs, by a graphic (ibid .) . Then, he showed how one can interpret trigonometric lines as the geometric construction of some imaginary quantity. Fo r
example, he showed, after having chosen lines a and b in a circle conveniently (AB = + b, BC = +c), that the tangents can be represented as th e
mean proportional i/-be (ibid ., 288) :
Verbi gram ; Si prorfum' ab A fumatur AB+b,
& prôrfum adhttc (in eaden reaa) BC = + c ; fitqueAC (=AB+BC-+b+c) diameter ci r
cull : erit fnus re&us-, feu media proportionalis,
BP=I/+bc.
Sinietrorfi n ab A (adeoque cum contrario figno )
fumatur -AB .----b; &, à B prorfum,BC= +c ;
rnanente cad= circuli diametro AC = - AB +BC
-b+ e : erit .Tangens, feu media proportionalis, EP-V-bc .
What was the effect of this textbook and in particular the impact of thi s
method for interpretation? I could find no trace of a discussion or a reception by contemporaries of Wallis or of later authors of the first half of the
eighteenth century. This silence is the more astonishing since Wallis ha d
expressed this approach in an earlier letter . In a letter of 6 May 1673, ad dressed to John Collins (1625-1683), (who was acting as secretary of th e
Royal Society), Wallis explained his interpretation of imaginary roots a s
"mean proportionals" in the context of a discussion on Cardan's rules fo r
solving higher degree equations .
As Wallis reported, he had earlier on had scruples as "too young an algebraist to innovate without example" . But since he had become "more
venturous" in the meantime, he "had several projects" for " designing geometrically" imaginary roots (Rigaud 1841, 578) . It seems that Collins did
not raise objections to Wallis's argumentation .
142
G. Schubring
MfM 46 : 2
It seems, therefore, that Wallis was the first to propose a geometric construction of imaginary quantities but that he remained without an echo and
an impact in his time, or even later on . Foncenex's discussion means a new
beginning of related reflections .
5 Wenceslaus J .G . Karste n
As a concluding element, I will present a little-known but highly instructive
example of a direct impact of Foncenex's mémoire, one which gives a muc h
more sophisticated geometric construction than those discussed up to now .
It is a paper on the logarithms of negative quantities, by W .J .G . Karste n
(1732-1787), a mathematics professor at one of the smallest German univer sities, at Bützow, when he published its first version in 1768 . Karsten was
later professor at Halle, then ranking as the second university in Germany ,
when he published a revised version in 1786 .
In this paper, which referred to Foncenex, Karsten, too, discussed the possi bility and legitimacy of a geometrical construction of imaginary quantities .
He asserted, firstly, that an algebraically impossible (i .e . imaginary or complex) quantity does not admit a geometrical construction . He continued
nevetheless his reflection, by introducing the distinction that if an algebraically impossible quantity is given in the form b \/-1 one can construc t
the possible factor b geometrically . Karsten remarked that it would be sufficient to bear in mind that the geometric quantity thus constructed is no t
identical with the quantity sought (Karsten 1786, 379) .
He went on to give, as an example, a particular relation between a hyperbol a
and a circle . Karsten defined an equilateral hyperbola by the equation x 2 -
y 2 = 1 and the related circle by x 2 + z 2 = 1, with y = z //-1 (ibid . ,
380, see figure 5) . In this case, for x = 0 the ordinate is imaginary : y =
f-/-1 = + W-1, and putting CE = CF = 1 yields y = ± CE \/- 1 an d
+CE = -y-/-1 ; in general, all values y of the hyperbola for x betwee n
+1 and -1 are imaginary. And Karsten explained that one is able in thi s
way to draw ("zeichnen") the respective ordinates for all these x : thes e
ordinates are in themselves ("an sich") possible quantities, but they appea r
as imaginary quantities here in the calculus since they cannot intersect th e
hyperbola and are hence imaginary, regarded as ordinates of the hyperbola .
Moreover :
MfM 46:2
143
Figure 5 : Fig. No. 46
All ordinates of this circle are imaginary ordinates of the hyperbola ,
since z = ± y N/-1 ; but also vice-versa, all ordinates of the hyperbol a
are imaginary ordinates of the circle, since y = z '/-1. As a consequence, the circle is an imaginary part of the hyperbola, as the hyper bola is an imaginary part of the circle . (Karsten 1786, 380-381 ; my
translation, G . S .) . 12
And Karsten, who was aware of the periodicity of the circle, even showe d
that there is an infinity of such correspondences (ibid ., 381 ff.).
Resuming his discussion, Karsten asserted that the geometric design of a n
algebraic formula can never lead to consequences which are different from
those obtained by algebraic methods (ibid ., 385) . Remarkably, by this point
Karsten had skipped over the distinction between `possible' and ìmpossible '
algebraic terms .
Cajori seems to have been the only one who has discussed Karsten's paper .
He commented in the following terms that he could see no reception :
It looks very much as if transactions of academies had been in som e
cases the safest places for the concealment of scientific articles fro m
the scientific public (Cajori 1913, 111) .
12I
am quoting here from the revised later version . Although profoundly revised, the
example of this construction with the hyperbola and the circle had remained unchange d
as compared to the first version (Karsten 1768, 96-98) .
144
G. Schubring
MfM 46 :2 ,
One has to add that Cajori did not know that Karsten had organized a sec ond printing of his paper in 1786, which certainly did not remain unnoticed .
Actually, we have not studied sufficiently the impact of the enormous num ber of foundational reflections by mathematicians who are not regarde d
as great mathematicians on the evolution of the thinking of the greater
mathematical community. 13 The foundational work of marginal contributors prepared well the passage from an epistemology favouring a geometrical
relation to real-world existence to an epistemology favouring internal systemic coherence . It is quite paradoxical that this passage paved its way b y
reinforcing the geometrical legitimation of abstract mathematical objects .
Foncenex ' s proposal to admit imaginary quantities only as auxiliary mean s
during the calculation process but not in the final solution provoked th e
eventual breakthrough of their general admission as algebraic concepts.
References
Jean le Rond d'Alembert, Opuscules Mathématiques. Tome Première, Sixième Mémoire ,
"Sur les Logarithmes des quantités négatives", Paris 1761, 180-230 .
Argand, Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques, Paris 1806 (imprimé anonymement, réédité en 1874 pa r
J . Hoüel) .
Argand, "Solution des deux problèmes proposés à la page 249 du III . e volume des Annales ,
avec quelques applications à la construction des thermomètres métalliques en form e
de montre", Annales de mathématiques pures et appliquées, 1813/14, 4, 29-41 .
[1813a]
Argand, "Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques", Annales de mathématiques pures et appliquées, 1813/14 ,
4, 133-148 . [1813b ]
Adrien-Quentin Buée, "Mémoire sur les Quantités imaginaires", Philosophical Transactions of the Royal Society of London. For the Year MDCCCVI .Part 1, London 1806 ,
23-88 .
13 0ne further indication for the wide diffusion of the idea of geometrical representatio n
is a review in the Göttingische gelehrte Anzeigen of 6 March 1806 by Tobias Mayer, physics
professor at the Göttingen University, where he asked for concrete modes of geometrica l
construction for imaginary roots . As a source, he pointed to a paper by Heinrich Kühn o f
1750 in the Petersburg Academy Transactions . Despite its promising title, it is a rathe r
confused paper since the author attributes to the same area of, say a square, a positive
or a negative value - depending on in which quadrant it is situated - and deduces from
negative area values that the corresponding side lengths are imaginary (Kuehn 1753) .
MfM 46 :2
145
Florian Cajori, " History of the exponential and logarithmic concepts", American Mathematical Monthly, 1913, 20, 107-117 .
Lazare Carnot, Géométrie de Position, Paris an XI (1803) .
Augustin Louis Cauchy, "Mémoire sur les quantités géométriques" (1847), OEuvre s
Complètes, IIe Série, Tome XIV, Paris : Gauthier-Villars, 1938, 175-202 .
Leonhard Euler, Opera Omnia. series Quarta A : Commercium Epistolicum . Volume n
Quintum : Commercium Epistolicum cum A . C. Clairaut, J . D'Alembert et J. L .
Lagrange.. Ed. Adolf P. Youschkevitch et René Taton (Basel : Birkhäuser, 1980) .
François Daviet de Foncenex, "Refléxions sur les quantités imaginaires", Miscellanea
Philosoph'ico-Mathematica Societatis Privatae Taurinensis, 1759, Tomus Primus ,
113-146 .
François Daviet de Foncenex, "Eclaircissemens pour le Mémoire sur les quantités imaginaires inséré dans le premier Volume" , Mélanges de Philosophie et de Mathématiqu e
de la Société Royale de Turin pour les Années 1760-1761 [Tome II], 337-344 .
Bernard le Bovier de Fontenelle, Elements de la Géométrie de l'Infini, Paris 1727 .
Jacques Frédéric Français, " Nouveaux principes de Géométrie de position, et interprétation des symboles imaginaires " , Annales de mathématiques pures et appliquées, 1813/14, 4, 61-72 . [1813a]
Jacques Frédéric Français, "Lettre sur la théorie des quantités imaginaires", Annales d e
mathématiques pures et appliquées, 1813/14, 4, 222-230 . [1813b ]
Carl Friedrich Gauß, "Theoriam Residuorum Biquadraticorum . Commentatio secunda" ,
Comm,entationes societatis regiae scientiarum Gottingensis recentiores, 1832, VII
= Carl Friedrich Gauf.1 Werke, Band II (Göttingen 1863), 95-148 .
Carl Friedrich Gauß, "Anzeige : Theoriam Residuorum Biquadraticorum . Commentatio secunda", Göttingische gelehrte Anzeigen, 23 . 4 . 1831 = Carl Friedrich Gauf.? Werke,
Band II (Göttingen 1863), 169-178 .
Hermann lankel, Theorie der complexen Zahlensysteme, insbesondere der gemeinen imaginären Zahlen und der Hamiltonschen Quaternionen nebst ihrer geometrische n
Darstellung (Leipzig: Voss, 1867) .
Jules Hoüel, "Avertissement de l'editeur" . In : Argand, 1874, v-xix .
Wenceslaus J . G . Karsten, "Abhandlung von den Logarithmen verneinter Grössen" , Abhandlungen der Churfürstlich-baierischen Akademie der Wissenschaften (München ,
1768), 1-108 .
Wenceslaus J . G . Karsten, "Von den Logarithmen der verneinten und unmögliche n
Grössen", Mathematische Abhandlungen, theils durch eine Preisfrage . . . , theils
durch andere neuere Untersuchungen veranlasst (Halle : Renger, 1786), 285-390 .
Henricus Kuehn, "Meditationes de quantitatibus imaginariis construendis et radici s
imaginariis exhibendis", Novi Commentarii Academiae Scientiarum Imperiali s
Petropolitanae, Tom. III ad Annum MDCCL et MDCCLI, Petersburg, 1753, 170223, Tab . I-III .
Herbert Pieper, Die komplexen Zahlen . Theorie - Praxis - Geschichte (Berlin : Deutscher
Verlag der Wissenschaften, ].984) .
Robin E . Rider, A Bibliography of Early Modern Algebra. Berkeley papers in History o f
Science, no . VII (Berkeley : University of California, 1982) .
Stephen Jordan Rigaud, Correspondence of scientific men of the seventeenth century. Vol .
II (Oxford, 1841) .
146
G. Schubring
MfM 46 :2 .
Schubring, Gert : "Ruptures dans le statut des nombres négatifs", petit x (Grénoble), no .
12, 1986, 5-32 .
H . Valentiner, "Première préface" . In : Wessel 1897, iii-x .
John Wallis, De Algebra Tractatus ; Historicus et Practicus . Operum Mathematicorum
Volumen alterum (Oxford : Sheldon, 1693) .
Wessel, Caspar: Essaz sur la représentation analytique de la direction . Traduction du
mémoire intitulé: Om Directionens analytiske Betegning. Préfaces de H . Valentine r
et T .-N . Thiele, Copenhague 1897 . English translation : On the Analytical Representation of Direction . An Attempt Applied Chiefly to Solving Plane and Spherica l
Polygons . Translated by Flemming Damhus . With introductory chapters by Bodi l
Branner, Nils Voje Johansen, and Kirsti Andersen . (ed . Bodil Branner and Jesper Lützen), Matematisk-fysiske Meddelelser 46 :1, Det Kongelige Danske, Viden skabernes Selskab, Copenhagen 1999 .
MfM 46:2
14 7
Inexplicable? The status of complex numbers in
Britain, 1750-1850
Adrian Rice *
"The history of algebra shows us that nothing is more un sound than the rejection of any method which naturall y
arises, on account of one or more apparently valid cases i n
which such method leads to erroneous results . Such cases
should indeed teach caution, but not rejection : if the latte r
had been preferred to the former, negative quantities, an d
still more their square roots, would have been an effectual
bar to the progress of algebra, which would have been con fined to that universal arithmetic of which Newton wishe d
it to bear the name : and those immense fields of analysi s
. would have been not so much as discovered, much less
cultivated and settled ." [De Morgan 1842b, 566]
1 Introductio n
The period from the mid-18th to the mid-19th century is generally regarde d
as the time when complex numbers were finally accepted as legitimate algebraic objects by the mathematical community . It is also widely viewed tha t
it was the work of Carl Friedrich Gauss (1777-1855) which was primaril y
responsible for bringing these numbers into the mathematical mainstream ,
with his proofs (in 1799, 1815, 1816 and 1849) of the fundamental theore m
of algebra, as well as his description of their geometrical representation i n
* Department of Mathematics, Randolph-Macon College, Ashland, VA 23005-5505 ,
U .S .A .
148
A . Rice
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1831, serving as the final impetus to their eventual acceptance . [Lewis 1994 ,
725]
However, while this is basically true, especially for countries at the hear t
of the mathematical mainland of western Europe, such as France and Ger many, the story of the acceptance of complex numbers was a little different in
the British Isles.' So, although the general recognition of complex numbers
dates from around the same time in Britain as on the rest of the continent ,
there were noticeable differences in the underlying causes . One of these was
the relatively isolated position of British mathematicians arising from thei r
rejection of the continental approach to the calculus and related topics . 2
Thus, while still in touch with developments in mainland Europe, mathematicians in Britain were inevitably slower in assimilating results from th e
continent than might otherwise have been the case .
The country's peripheral geographical position, together with the effects o f
a prolonged war with France, also resulted in the slight delay and limite d
availability of continental publications (especially obscure ones) reachin g
its shores . Consequently, if this paper neglects to draw detailed attentio n
to the work of, for instance, Wessel, Argand or Grassmann, it is becaus e
their British contemporaries would have known little or nothing of them although in this regard, they were far from unique . For example, it shoul d
be said that in cases such as that of Grassmann, whose work was almos t
unintelligible to contemporary mathematicians, many of his compatriot s
were equally unacquainted with his work . [Lewis 1977]
One final factor was the distinctive intellectual environment in Britain a t
this time, with its emphasis on wide-ranging amateurism rather than professional specialisation, in which mathematical questions were often intimatel y
connected to more general philosophical problems . Thus the British debat e
over complex numbers has been interpreted as part of "a three-quartersof-a-century dialogue on general terms and sound reasoning in which major British thinkers intricately interwove mathematical and philosophica l
insights" . [Pycior 1984, 438] In this view, the change in predominant philosophy among British scientists in this period, from Locke to Berkeley t o
'By this, as well as by the word "Britain", I refer to the countries of England, Wales ,
Scotland and Ireland . The latter was officially incorporated as part of Britain by the Ac t
of Union in 1801, although it had been effectively under British rule for centuries .
2 Although this isolation was not as complete as earlier historians have suggested .
[Guicciardini 1989, vii--viii]
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The status of complex numbers in Britain, 1750-1850
149
Kant, resulted in a corresponding shift in mathematical point of view to wards, among other things, complex numbers .
The purpose of the present article is to survey these changing British at titudes during this time, a subject which, while it has been the subject o f
much recent research, still awaits a definitive history . It will therefore proceed as follows . Following a brief summary of the early history of comple x
numbers (j2), it will review the peculiarly British position towards the m
in the period up to around 1830, in order to determine precisely how the y
eventually came to be accepted in Britain 69-4) . Sections 5 and 6 will the n
highlight the largely forgotten but influential work on imaginary number s
of two British mathematicians, John Warren and John Thomas Graves ,
the details of which have been essentially overlooked by recent historian s
of mathematics . The influence of this work is traced in §7, via the familiar
story of the birth of quaternions, concluding with a survey of how the notio n
of "inexplicable quantities" subsequently came to be extended .
2 The first 200 year s
The early history of imaginary and complex numbers is well documented .
Their first recorded use is by Girolamo Cardano (1501-1576) in his Ars
Magna of 1545 . There he poses the problem of dividing 10 into two parts
such that their product is 40, obtaining the solutions 5 + ./-15 and 5 \/-15 . However, his attitude towards them was hardly progressive . Indeed ,
given contemporary reservations about the validity of negative numbers in
algebra, it is hardly surprising that Cardano describes the use of these new
forms as involving "mental tortures" [Cardano 1968, 219] .
It was Descartes who, in La Géométrie (1637), introduced the term "imaginary" [Smith and Latham 1954, 175] to describe such entities, 3 although he
himself did not regard them as actual numbers . From the mid-17th centur y
onward, however, mathematicians appear to have had fewer qualms abou t
imaginaries and made frequent use of them . Newton believed that complex
roots were helpful for determining which problems have real or explicabl e
solutions . He said : "it is just that the Roots of Equations should be ofte n
3 Albert Girard, in 1629, referred to them as "solutions impossibles" - the phras e
"impossible numbers" was widely employed during the eighteenth and early-nineteent h
centuries . It was Gauss who introduced the term "complex" in 1831 . [Gauss 1876, 1021
150
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A . Rice
impossible, lest they should exhibit the cases of Problems that are impossible as if they were possible ." [Newton 1728, 193] Yet their epistemologica l
status remained unclear, as reflected in Leibniz's description of them a s
"that amphibian between being and not-being, which we call the imaginar y
root of negative unity" [Leibniz 1858, 357] .
Not surprisingly, considering their ambivalent status, by the early 18th century, results concerning negative and complex numbers were the cause o f
heated debates among many of the leading mathematicians . One of the mos t
vigorous discussions concerned the nature of their logarithms. [Cajori 1913 ,
35-47, 75-84] During this period, Leibniz and the Swiss mathematician Johann Bernoulli put forward contrasting opinions on the subject, the latte r
proving for example that
7r=
2log \/- 1
-\/- 1
Related results soon followed which quickly established a connection between logarithms, exponentials and trigonometric functions . In 1714, the
Englishman Roger Cotes (1682-1716) was able to show that
x-/-1 = log(cos x + ,i-1 sin x) ,
which of course is equivalent to Euler's later result
e x`V-1 = cos x
+ A/-1 sin x.
Another theorem which originated in England at this time was the following ,
which Euler was also able to re-formulate and prove :
(cos x ± 'V-1 sin x)n =
cos nx ± //- 1
sin nx, n E R .
This result is implicitly used in the work of the French emigré Abraham
de Moivre (1667-1754), although he never stated it or attempted a proof .
[Schneider 1968]
The question of imaginary logarithms was finally resolved by Euler in 1749 .
By writing a complex number x = a + bV-1 as r(cos 0 + \/-1 sin 0), where
r = e°, he obtained
x = e c (cos 0 +
sine) = ece
( 0+2 70 .
71
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15 1
Thus
log x = c + -V-1(0 ± 2n7r), nEN orn =0.
He therefore concluded that, when x is positive and real, one value of th e
logarithm will be real, but for negative and imaginary values of x, all logarithms will be imaginary. [Euler 1749] But despite this successful resolution, no one was any clearer as to what complex numbers actually were, o r
whether it was mathematically valid to use them .
Moreover, because of the anti-intuitive nature of the results obtained b y
their use, even those who admitted their value in mathematical researc h
were liable to make mistakes . For example, although Euler correctly calculated that = e -7/2
0.20787957 . . . , he omitted to mention the
other values this number could be . Another error, made by Bernoulli an d
D'Alembert, was that (-a) 2 = a2 implied log(-a) 2 = log a 2 . This led them
to the result that 2log(-a) = 2 log a, or log(-a) = log a . [Woodhouse 1801 ,
114]
The resulting mixture of correct and erroneous conclusions concerning negative and complex numbers served only to intensify the confusion . As late a s
1815, the British mathematician Charles Hutton reported that "the arithmetic of these imaginary quantities has not yet been generally agreed on "
[Hutton 1815, I, 675], contrasting Euler's claim that \/-2 x '/-3 =
with William Emerson's 4 belief that '/-a x '/-b = \/-ab . "And thus," h e
concluded, "most of the writers on this part of algebra are pretty equall y
divided ." [Hutton 1815, I, 675]
3 The British debate 1750-183 0
The unease with negative and imaginary numbers was especially strong i n
Britain . [Pycior 1997] The primary concern was that the lack of a proper def inition of negative numbers, and hence imaginary quantities, rendered the m
meaningless . John Wallis (1616-1703), for example, in his Arithmetica Infinitorum (1655) had expressed the view that negatives were not only greate r
4 Emerson (1701-1782), an English mathematician, was later described as "the write r
of many works, which had considerable celebrity : but he was as much overrated as Thoma s
Simpson was underrated" [De Morgan 1847, 78] .
152
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than zero but actually larger than infinity, a view also held by Euler ove r
a century later . [Kline 1972, 253, 593] Yet, throughout the 18th century ,
negative numbers were also variously defined as "quantities less than nothing" and "quantities obtained by the subtraction of a greater quantity fro m
a lesser" . [Pycior 1981, 28] Since it was almost universally acknowledged
that there existed no adequate justification for either negatives or comple x
numbers, it is hardly surprising that opposition to their employment wa s
eventually expressed .
This came in the form of A dissertation on the use of the negative sig n
in algebra, published in 1758 by Francis Maseres (1731-1824), a Fellow o f
Clare College, Cambridge and the Royal Society of London . In this wor k
he advocated the complete rejection of all negative numbers as solutions
to equations or algebraic processes . His purpose, as he explained in the
preface, was to "remove from some of the less abstruse parts of Algebra ,
the difficulties that have arisen therein from the too extensive use of the
Negative Sign, and to explain them, without considering the Negative Sig n
in any other light than as the mark of the subtraction of a lesser quantit y
from a greater" . [Maseres 1758, i]
He went on to demonstrate how to evade such difficulties by separatin g
those equations with negative roots from the rest and only admitting thei r
positive roots . In Maseres' algebra, therefore, there was no place for negative
(and therefore also imaginary) roots to any form of equation since "they
serve only, as far as I am able to judge, to puzzle the whole doctrine o f
equations, and to render obscure and mysterious things that are in thei r
own nature exceeding plain and simple . . . " In his opinion, "It were to b e
wished therefore that negative roots had never been admitted into algebra ,
or were again discarded from it" . [Maseres 1758, 34 ]
Although the Dissertation was somewhat radical in its emphatic denial o f
the validity of negative concepts, and although Nlaseres' extreme opinion s
were not shared by the majority of his contemporaries, it should not b e
assumed that he or his work were ignored or derided . On the contrary,
Maseres was a competent and respected mathematician who, two years later ,
would compete (albeit unsuccessfully) for the prestigious Lucasian chair o f
mathematics at Cambridge . [Pycior 1997, 306-7] Indeed, the fact that a
man of his intellectual stature was prepared to make such a bold statement
added impetus to the need felt among the British to justify the obscur e
notions of negatives and imaginaries .
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15 3
Towards the end of the century, Maseres was joined in his attack of thes e
"impossible" quantities by another Cambridge mathematician, Willia m
Frend (1757-1841), 5 whose Principles of algebra, published in two volume s
in 1796 and 1799, endorsed the abridged interpretation of the subject . In the
Principles, Frend criticised foregoing algebra texts, including both Newton' s
Arithmetica Universalis (1707) and Maclaurin's Treatise of Algebra (1742) ,
for attempting to define negatives by resorting to analogy, such as financia l
debts or directed lines : "when a person cannot explain the principles of a
science without reference to metaphor, the probability is, that he has neve r
thought accurately upon the subject" . [Fend 1796, x]
Supporting Maseres ' contention that negative and imaginary numbers wer e
"a parcel of algebraick quantities, of which our understandings cannot for m
any idea" [Maseres 1778, 947], Fend branded such concepts as "jargon ,
at which common sense recoils", adding that "like many other figments, i t
finds the most strenuous supporters among those who love to take things
upon trust, and hate the labour of a serious thought" . [Frend 1796, xi] His
chief concern was that if these poorly-defined concepts were used in algebra ,
even valid reasoning could produce incorrect results : "from false notions ,
falsehood must necessarily flow, if the reasoning employed upon them ha s
been properly conducted" . [Frend 1798, 3]
As a mark of his endorsement of Fend's work, Maseres provided an appendix on the solution of cubic and quartic equations . Here he observe d
that the treatment of this subject by previous authors "has been made th e
subject of much mysterious and fantastick reasoning, (or, perhaps, I ough t
rather to say, discoursing, since it deserves not to be called reasoning,) concerning negative and impossible quantities . . . All these writers have . . .
treated this subject with an astonishing degree of obscurity, and almost a s
if they had been contending with each other which should treat it mos t
obscurely" . [Frend 1796, 253-4] In consequence, he claimed, "the Science o f
Algebra, or Universal Arithmetick, has been disgraced and rendered obscur e
and difficult, and disgusting to men of a just taste for accurate reasoning" .
[1Vlaseres 1800, 1v ]
While Maseres and Frend may have been in the minority with their belief i n
the necessity of excluding negatives and imaginaries from algebra, they wer e
very much in harmony with the majority of the mathematical communit y
5 For a biography of Frend, see Knight 1971 .
154
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in their diagnosis of the problem . In the 75 years following Maseres' initia l
attack, a number of works by prominent British mathematicians responde d
to the anti-negative argument . Indeed, the last half of the 18th century an d
the opening third of the 19th saw the question of negative and imaginar y
numbers occupy a major place in the discussions of British mathematicians ,
philosophers and men of science . However, the overall consensus was that ,
although a satisfactory definition was still absent, their utility as indispensable mathematical tools meant that they could not be abandoned .
One of the first to come to their defence was the Scottish mathematicia n
John Playfair (1748-1819) who, in a paper of 1778, presented his reader s
with the following paradox : "If the operations of this imaginary arithmeti c
are unintelligible, why are they not also useless?" [Playfair 1778, 321] Hi s
answer was that although imaginary numbers, and the operations performe d
on them, have no direct meaning themselves, they act as "notes of referenc e
to others which are significant" . [Playfair 1778, 326] Thus, although the re sults obtained through them are justified by argument from analogy, thi s
does not impair their validity . Rather, the use of negatives and imaginarie s
serves as a means of deducing meaningful results when no other method i s
available . "For this reason, many researches concerning it, which in them selves might be deemed absurd, are nevertheless not destitute of utility ."
[Playfair 1778, 342]
In the following decade, William Greenfield, professor of rhetoric at the Uni versity of Edinburgh, entered the controversy . Although he conceded that
"the Method of negative quantities . . . is supported, rather by inductio n
and analogy, than by mathematical demonstration," [Greenfield 1788, 134 5] he was opposed to the idea of their exclusion from mathematics . While
agreeing that "the whole business of algebra might be carried on withou t
the consideration of the negative roots," he stressed that since their ad mission "evidently affords so great elegance and universality to algebraica l
solutions ; to find our author [1laseres] gravely declaring that he can se e
no advantage in it, is perfectly astonishing" . [Greenfield 1788, 136] Indeed ,
Greenfield goes on to rebuke Maseres for failing to "exert his industry an d
ingenuity, rather to confirm than to destroy ; rather to demonstrate, ho w
far we might rely on the method of negative quantities, than to overturn a t
once so great a part of the labours of the modern algebraists" . [Greenfiel d
1788, 136]
The debate continued into the 19th century with two works by a young
MfM 46 :2
15 5
Cambridge mathematician, Robert Woodhouse (1773-1827) ; a paper published by the Royal Society in 1801 and a textbook entitled The Principles of
Analytical Calculation which followed two years later . In the first of these ,
Woodhouse attempted to demonstrate why results obtained by means o f
imaginary numbers were necessarily true . To do this, he discarded Playfair's appeal to analogy, preferring to justify imaginaries by stressing th e
validity of their formal laws of operation as opposed to the meaning of th e
individual characters operated upon . As he argued, "a conclusion expresse d
by certain characters and signs, if general, must be true in each particula r
case that presents itself, on assigning specific values to the signs " . [Woodhouse 1801, 93 ]
Thus, he asserted that since (a+b) (c+d) and ac+ad+bc+bd are algebraically
equivalent for all real a, b, c and d, replacing the real quantities b and d with
the imaginary numbers bV-1 and d-/-1 would lead to the conclusion tha t
(a + b-/-1)(c + d \/-1) and ac + ad \/-1 + cb\/-1 - bd are also equivalent
forms . But, he said, these latter expressions are "not proved equivalent, but
put so, by extending the rule demonstrated for the signs of real quantitie s
to characters that are insignificant" . [Woodhouse 1801, 93] Similarly, he
claimed, just as (a+b) x was regarded as being equivalent to ax +xax-l b+ . . .
so (a+b)x 3 -1 was likewise equivalent to ax'V-l +x~/-lax -1 1 b+ . . . "no t
by any proof, but by the extension of a rule" . [Woodhouse 1803, 8-9 ]
Thus, if either negative or imaginary quantities were to be made the subjects of mathematical demonstration, "it must be in consequence of som e
arbitrary rule" . [Woodhouse 1803, 7] But since these rules had already bee n
established for general terms, regardless of their meaning, he could therefor e
conclude that "although the symbol \/-1 be beyond the power of arithmetical computation, the operations in which it is introduced are intelligible, an d
deserve, if any operations do, the name of reasoning" . 6 [Woodhouse 1801 ,
108]
An alternative defence was offered by the French expatriate Adrien-Quenti n
Buée in 1806. Unlike Woodhouse, however, he emphasised the need fo r
meaning in algebra . Dividing it into "universal arithmetic" and a "mathe6 It is interesting to note that, despite his vigorous defence of the use of comple x
numbers in algebra, in a letter to Maseres, dated November 16, 1801, Woodhouse writes :
"Till the doctrines of negative and imaginary quantities are better taught than they ar e
at present taught in the University of Cambridge, I agree with you that they had bette r
not be taught" . [De Morgan 1842a, 147]
156
A.
Rice
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matical language", where the symbols 1- and - were also characterised b y
the vaguely-defined notion of "quality", Buée returned to Playfair's philosophy of analogy to make his case . Thus "when one says that a negative
quantity is smaller than zero . . . it is not the quantity which is smaller than
zero ; it is the quality which is inferior to nullity . For example, if my debt s
exceed my assets, I am poorer than if I had neither assets nor debts" . [Buée
1806, 25] While this paper contained some original ideas on the meaning an d
interpretation of both negative and imaginary quantities (including geometric representation), it was "presented in a very vague and unscientific for m
. . . [and] accompanied by many other attempts which were either imperfec t
or altogether erroneous" . [Peacock 1830, xxvii]
It was in a critique of Buée's memoir, published 30 years after his initia l
foray, that Playfair made a second contribution to the debate . Rejecting all
intervening explanations, he insisted that while the utility of imaginarie s
was certain, neither the numbers themselves nor the operations performe d
on them could be assigned any meaning . He illustrated his point by considering the operation of dividing by an imaginary number : "what is meant
by dividing by an impossible quantity, or telling how often an impossible
quantity contains another ; if the quantity be impossible, to multiply or divide by it, or to make it the subject of any arithmetical operation, must b e
impossible also . The operations performed with the symbols are therefor e
destitute of meaning ; they are as imaginary as the symbols themselves ."
[Playfair 1808, 315 ]
And so the debate rumbled on for the next twenty years, with additional
articles and textbooks adding further opinions to those already mentioned .
For the most part, however, these contained mainly reiterations or mino r
variations of views already aired [Bonnycastle 1820, II, 24-25n ; Hutton 1815,
I, 675, II, 93] . The final significant contribution came from Davies Gilber t
(1767-1839) in his valedictory address as President of the Royal Society i n
1830 . Although not a practising mathematician, Gilbert was able to offe r
some perceptive remarks on the current status of imaginaries in mathematics . He referred to them as "Creations merely of arbitrary definition ,
endowed with properties at the pleasure of him that defines them" . [Gilbert
1831, 93] Their use in mathematics is thus perfectly justified since, althoug h
an imaginary quantity has only potential existence while the V-1-symbol i s
attached to the -1, "it stands ready to exist in energy whenever that sig n
is removed" . [Gilbert 1831, 94]
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157
The year of Gilbert's address marks the end of three-quarters of a century o f
dispute over the status, meaning, interpretation and validity of negative an d
imaginary numbers . This was not because interest in the subject had wane d
or a consensus had been reached, but because of the publication of a ne w
work which offered a new approach to the practice of algebra and effectivel y
rendered the question of the legitimacy of complex numbers immaterial .
That work was A Treatise on Algebra by George Peacock (1791-1858), th e
first publication to propose a purely symbolic approach to the subject
although, as we shall see, some aspects of the new philosophy were no t
entirely novel .
4 Peacock's symbolism : a new approach ?
The algebraic work of George Peacock 7 represented the most significant
contribution to the (British) debate thus far . This is largely to be found i n
two main sources . The first was his aforementioned Treatise on Algebra, published in 1830, of which a second edition, substantially revised and extended ,
appeared in two volumes in 1842 and 1845 . His other principal publicatio n
was a book-length "Report on the recent progress and present state of certain branches of analysis", which he presented at the third meeting of th e
British Association for the Advancement of Science in 1833 . With these two
works, Peacock initiated a different, more abstract, algebraic methodology ,
which fundamentally altered the way the subject was perceived in Britain .
In common with recent authors on the subject, Peacock was in full agree ment with the claims of Maseres and Frend regarding the inadequate justification of negatives and imaginary numbers in mathematics . However, he was
also adamant that any restriction which limited algebra merely to a for m
of universal arithmetic would be equally unsatisfactory . This, he explained ,
was because of the "great multitude of algebraical results and propositions ,
of unquestionable value and of unquestionable consistency with each other ,
which were irreconcilable with such a system, or, at all events, not deducibl e
from it, . . . and which made it necessary to consider negative and even im possible quantities as having a real existence in algebra, however vain migh t
be the attempt to interpret their meaning" . [Peacock 1833, 190-1 ]
' Like Maseres, Frend and Woodhouse, Peacock was a Cambridge man, being a fello w
and tutor of Trinity College, Cambridge, and later the University's Lowndean professo r
of mathematics .
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The root of the problem, he decided, was that "Algebra has always bee n
considered as merely such a modification of Arithmetic as arose from th e
use of symbolical language, and the operations of one science have bee n
transferred to the other without any statement of an extension of thei r
meaning and application" . [Peacock 1830, vi] This had led to the erroneous
practice of using arithmetical symbols in algebra as if they represented th e
same operations with fewer limitations . The result was that, by appealing
to a basis in arithmetic, mathematicians had enlarged the scope of thes e
operations without adequate justification . In other words, they had taken a
special case and used it to justify a generalisation .
Peacock's solution was to divide algebra into two distinct subjects, arithmetical and symbolical . Arithmetical algebra was basically the system o f
universal arithmetic insisted upon by Maseres and Frend . Here, symbol s
could only represent positive real numbers, "whether abstract or concrete ,
whole or fractional, and the operations to which they are subject are assumed to be identical in meaning and extent with the operations of th e
same name in common Arithmetic" . [Peacock 1842-5, II, 1] Symbolical algebra, on the other hand, was a far more abstract system "which regard s
the combinations of signs and symbols only, according to determinate laws ,
which are altogether independent of the specific values of the symbols them selves" . [Peacock 1830, vi ]
In this second system, the symbols could be not only "the general representatives of numbers, but of every species of quantity" . [Peacock 1830, ix]
Moreover, it was now possible to assume that operations such as subtractio n
or extracting square roots were possible in all cases, thus enabling negativ e
and imaginary quantities to be perfectly legitimate within this system . The
emphasis now, then, was not on what symbols such as -a or i/-1 actually
meant but under what laws they operated . Thus, for Peacock, symbolical
algebra was a subject based not on a generalisation of arithmetical concepts ,
but on a series of arbitrary assumptions envisaged as "a means of evadin g
a difficulty which results from the application of arithmetical operation s
to general symbols" . [Peacock 1830, ix] In short, given that the questio n
posed was to legitimise negatives and imaginaries in algebra, Peacock ha d
answered, not by attempting to clarify the meaning of such entities, but b y
redefining what was meant by algebra itself .
Yet, although Peacock's views and ideology were certainly different to wha t
had gone before, it has been successfully argued that he was probably influ-
MIM 46:2
159
enced by previous (unpublished) work by Charles Babbage [Dubbey 1977 ;
1978, 93-130], and that both Peacock and Babbage can be seen to be building on ideas previously laid down by Woodhouse [Becher 1980 ; Nagel 1935 ,
446] . In turn, Woodhouse can be said to owe an intellectual debt to th e
writings of George Berkeley, while Babbage was deeply affected by the philosophy of Dugald Stewart . [Pycior 1984, 434-38 ; Sherry 1991, 47-54] In deed, looking back, one can even see the germ of a symbolic philosophy i n
Playfair's paper of 1778 : "In algebra again every magnitude being denote d
by an artificial symbol, to which it has no resemblance, is liable, on som e
occasions, to be neglected, while the symbol may become the sole object o f
attention ." [Playfair 1778, 319 ]
In his separation of algebra into two distinct parts, Peacock's approac h
also displays the influence of Buée's paper of 1806 . Indeed, by virtue of hi s
earlier division of algebra into "universal arithmetic" and a more general
"mathematical language", Buée was later recognised as being "though in a n
imperfect manner, . . . the first formal maintainer of that exposition which
remove[d] the long standing difficulty [of negative and imaginary numbers]" .
[De Morgan 1842a, 151] But Buée's work, flawed though it undoubtedly was ,
exercised influence not only on the innovative work of Peacock with regar d
to algebra, but also on a different attempt to justify imaginary numbers ,
this time by means of geometry.
5 Warren and the geometrical approac h
The earliest hint at the possibility of a two-dimensional representation o f
complex numbers was in John Wallis's Treatise of Algebra of 1685 . There ,
he had demonstrated a procedure for determining the real coefficients of th e
complex roots of quadratic equations, by means of two perpendicular axes ,
one for the reals and one for the imaginaries - although this latter was no t
necessarily the y-axis . [Wallis 1685, 266-73] However, while correct, Wallis' s
method, and others formulated during the 18th century, failed either to exer t
any influence on further developments or to persuade the mathematica l
community of the validity of complex numbers .
Thus, despite its great merit, when the Norwegian Caspar Wessel (1745 1818) published his paper "On the analytic representation of direction "
in 1799, it received virtually no attention until its republication nearly a
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hundred years later . A similar fate befell the work of Jean-Robert Argand .
While his Essai sur une manière de représenter les quantités imaginaire s
dans les constructions géométriques (1806) did provoke some discussion o n
the subject in the French journal Annales des mathématiques around 181315, its immediate effect was otherwise entirely negligible . 8
Not surprisingly, therefore, neither Wessel's paper nor Argand's Essai had
any influence on the debate in Britain . Indeed, it is almost certain that none
but a tiny fraction of British mathematicians were aware of the existenc e
of either work, and still fewer had actually read them . 9 In Britain, it was
Buée's paper (coincidentally published in the same year as Argand's work )
which appears to have been the first to offer a new approach to the geometrical representation of complex numbers . Like Argand, Buée interprete d
the symbol A/-1 as representing perpendicularity in geometry . Thus passing from a point x on the real axis to a point -x on the same line coul d
be achieved by rotating the point through a right angle and then repeatin g
the process, which would be equivalent to multiplying x by -/-1 and the n
again by '/-1, to obtain -x.
However, as has been mentioned, the clarity of the explanation and th e
rigor of some of Buée's arguments left a good deal to be desired, bein g
accompanied by much that was unintelligible or simply wrong, such as a
"proof" that (-/-1) 'n =
[Buée 1806, 67-71] Indeed, in the preface
to his Treatise, Peacock pointed out that "even in those cases in which hi s
conclusions were correct, his reasonings were insufficient to establish them" .
Nevertheless, he still acknowledged the influence of the paper " as having firs t
directed my attention to this subject" . [Peacock 1830, xxvii]
It would appear, however, that Buée's influence did not extend much further ,
since the next significant work to be published on the subject made n o
reference to him (or indeed any other previous writers on the subject) a t
all . This work, A treatise on the geometrical representation of the square
roots of negative quantities (1828) was written by John Warren (1796-1852) ,
yet another Cambridge mathematician, this time from Jesus College .'° Th e
8 See the paper by Gert Schubring in this volume .
9 De Morgan listed Argand's Essai among the works on this topic "which I have eithe r
not seen, or cannot immediately obtain" [De Morgan 1849, vi], while Hamilton admitte d
that he knew of it "only by Dr . Peacock's mention of it in his Report [of 18331" [Hamilto n
1853, 31 ]
'
°Warren's explanation for the absence of any citation to Buée was that he was "not
MfM 46:2
16 1
book was original in many ways, not least in its notation . Warren expresse d
a line of length a, inclined to the real line at an angle A (< 360 0 ), as ô . More
generally, pa denoted a line of length a inclined at an angle of A+p .360° . So
that, "if (, ) m/n = b, b is inclined to unity at an angle = n (A + p.360°) . "
[Warren 1828, 29]
Using this convention, Warren was able to prove that both roots of - 1
represented anticlockwise rotations of 90° and 270°, respectively:
1 has two values, viz . (01)1/2 and 11 ) 1/2 '
Now -1 is inclined to unity at an angle = 180° ,
1 _1 ) 1/ 2
2(180°) = 90°
2
1/
and ( 11 ) 2
[Hence] if ( 01)1/2 be represented by
_
2 (180° + 360°) = 270°
\/- 1
(-11) 1 / 2 will be represented by [Warren 1828, 59-60]
Taking this concept further still, he demonstrated that if c = e(1) n, for
e > 0 and 0 < n < 1, then when n < 4 , c would be of the form +a+ b\/- l;
similarly, for 1,4 < n < , c = - a + b,/-l, and so on . [Warren 1828, 64]
Warren's Treatise thus gave the most comprehensive rationale to date (in
English) for the geometrical representation of complex numbers ; and while
there were omissions, 11 it was praised for having "completely succeeded in
giving an interpretation of the roots of unity, when attached to symbol s
which denote lines in Geometry" . [Peacock 1830, xxvii]
The work of John Warren, though virtually unheard of outside Britain, an d
largely forgotten within twenty years inside, was otherwise significant in
two respects . Firstly, it would later influence Hamilton in his research o n
algebraic triples and hence provide a starting point from which the theor y
of quaternions would develop . Secondly, it marks the genesis of a change i n
aware of the existence of M . Buée's paper till November 1827, when my treatise was i n
the press" . [Warren 1829a, 251 ]
11 The Treatise contained no interpretation of numbers of the form (a +
an oversight not realised by Warren until he read C .V . Mourey's La vraie théorie de s
quantités négatives et des quantités prétendues imaginaires (Paris, 1828) .
162
A . Rice
MfM 46 : 2
attitude amongst British mathematicians - an attitude completely share d
by Peacock - in which the justification of complex numbers is only part o f
a broader framework which is more concerned with how mathematics ca n
be extended once their use is permitted .
Indeed, if anything, Warren's work was less an attempt to justify "impossible" numbers than a bid to demonstrate their geometric utility . In fac t
he rarely used the word "impossible" to describe complex numbers, since
"by the word ìmpossible' no impossibility is necessarily implied, but on th e
contrary, . . . the quantities called impossible have a real existence, and ar e
capable of geometric representation" . [Warren 1829b, 340] This progressiv e
attitude characterised much of British work on complex numbers from thi s
point onwards . In British eyes, thanks to Warren and Peacock, the veracit y
of their use in both algebra and geometry was now established . 12 The question now was to discover in what ways the subject could be extended . Th e
first area considered was that of logarithms .
6 A new logarithmic controvers y
As was seen in §2, Euler had successfully solved the contentious problem o f
imaginary logarithms in 1749 . This had shown that the general logarithm i s
dependent on an arbitrary non-negative integer . However, the late 1820s and
early 1830s saw a minor reprise of the controversy in Britain . It began with
research undertaken by a young Irish mathematician by the name of Joh n
Thomas Graves (1806-1870) . A lawyer by profession, Graves had studied
mathematics at Trinity College, Dublin, and while there had formed a strong
and lasting friendship with fellow student William Rowan Hamilton (1805 1865) . For the next forty years, their regular correspondence would act as a
mutual source of stimulation for each other's mathematical work .
In 1826, Graves obtained a result that in the most general case of a, logarith m
of a complex number taken to a complex base, two integers are necessar y
12 Not all British mathematicians were immediately convinced, however . Taking the example of Augustus De Morgan, Helena Pycior points out that he "seems to have full y
recognised only in the mid-1830s that the geometrical representation of the complex numbers made a sound basis for legitimation of the imaginaries and justified the symbolica l
approach itself" . [Pycior 1983, 220] For a discussion of the reservations expressed by D e
Morgan, Frend, William Whewell and others with regard to symbolic algebra, see Pycio r
1982 .
MfM 46:2
16 3
to ascertain the solution . After lengthy consultation with Hamilton, Grave s
submitted the result to the Royal Society for publication in their Philosophical Transactions in 1828, but the referees (who included George Peacock )
remained unconvinced by the proof. The problem was that Graves' argument was long, circuitous and very difficult to follow . Had it not been for the
intervention of Hamilton on his friend's behalf, the paper would certainl y
have been rejected . [Hankins 1980, 262] Thanks to his persuasion, it finall y
appeared in 1829 under the title "An attempt to rectify the inaccuracy o f
some logarithmic formulæ" .
The crux of Graves' argument ran as follows : Starting from the premises
f (0) = cos 0 + v/-1 sin 0
f(0)=1+ 3-1 .0+ - + l 2 .8~ Th +
f (x0) = a value of If ( 0 )}x
f-1 f (0) = 2nnr + 0
f {x(2nir + 0)} = { f (0)}x , where n is an integer, 13
(1)
(2 )
(3)
(4)
(5)
he let y = a' and
a = f (0) .
(6)
Therefore, by (5) ,
ax = { f (0)}x = f{x(2n7r + 0)} .
But, by (4) and (6),
2n7 + 0= f-1 f(0) = f -1 (a) .
Therefore
y = ax
= f(xf-1(a))•
(7)
Re-arranging gave
log a y
loge
13
=
= f
f'(Y) 1 4
-1 (a)
(8)
Graves credits premises (1) and (2) to Euler, (3) and (4) to de Moivre, and (5) to
Louis Poinsot (1777-1859), although he admits that his knowledge of the latter's researc h
"is derived not from the original Essays, but from abstracts of their contents given in th e
Dublin Philosophical Journal" . [Graves 1829, 183 ]
164
MfM 46 : 2
A . Rice
From (2),
.f(-,/-1)=++ . . .+
1
1 .2 . . . n
+ . . . =e.
Therefore, by (4),
r1(e) 2nnr - \/-1 .
So the logarithms of
k2
(positive) were given b y
log e (k2) = f-1 (k 2 )
(9 )
1
2nir -
Then, by a series of elaborate substitutions, Maclaurin expansions and fur ther manipulation, Graves arrived at the expression
f - 1 (k2 ) = 2m7r + 2~/-1
11-k 2
1 + k2
. .
+ .+
1
1-k2 2nF 1
2n + 1 (1 + k 2 )
+ . ..
where m is an integer, which equals 2m~r when k 2 = 1. Thus, he said, by
(9), if k 2 = 1, the logarithms of unity will be given by 2n /1 .
The announcement of this result provoked a series of publications both sup porting and opposing it . Overseas, work by the French and German mathematicians Alexandre Vincent and Martin Ohm arrived at the same conclusion . 15 At home, however, Peacock, in his Report to the British Associatio n
of 1833, continued to express his doubts about its validity [Peacock 1833 ,
266-7n; Cajori 1913, 178] Support came once again from Hamilton who ,
at the Association's meeting in Edinburgh the following year, defended hi s
friend's theory, offering a proof which confirmed the expression Graves ha d
obtained . [Hamilton 1834 ; Hankins 1980, 391-6]
Objections still lingered, however, although less about the final result than
the methods Graves had used to obtain it . Doubts as to the soundnes s
14 Note that, by (4), there are an infinite number of possible values here, since f -l (y) =
f -1 f (x f -1 (a)) = 2nn + x f -1 (a) . We must assume that Graves is considering the cas e
where n = 0 .
15 Vincent, a professor at the Royal College at Reims, published results in 1832 whic h
corroborated those obtained by Graves . [Vincent 1832] The results of Ohm (a professor
at Berlin), which actually pre-dated those of Graves, were published in volume two o f
his Versuch eines vollkommen consequenten Systems der Mathematik in 1823 ; althoug h
British mathematicians only learned of them via the second edition . [Ohm 1829 ]
MfM 46:2
165
of the proof were expressed by Augustus De Morgan (1806-1871), professor of mathematics at University College London, in 1836. Pointing out
that Graves had used the premise f (0) = cos 0 + \/-1 sin O to prove tha t
{ f (0)} x = f (x f -1 (a)), De Morgan observed that "this is equally true wher e
f (0) = c c °, c having any value whatever, as appears sufficiently if the com mon processes of Algebra be true" . [De Morgan 1836a, 62n]
Graves responded by issuing an alternative (and much shorter) proof, thi s
2m, r
time designed "to show that 1 is among the values of e 2n,r -,f'-f ." [Graves
1836, 281] This time his premises were the standard Maclaurin expansion s
for cos x and sin x, together with the assumptions that
x x2
=
x3
...
y
y2
y3
}{
x 1 + 1 + 1 .2 + 1 .2.3 +
{1 + 1 + 1 .2 + 1.2.3 +
2
3
1 + x+y + (x+y) + (x + y ) + . . .
1
1 .2
1 .2.3
.
that
(1 xl
exlx+
x2
x3
..
)
+ 1.2 + 1 .2 .3+
and that
l x = cos(2wx7r) + -/-1 sin(2wx7r), for w = 0, +1, +2, +3, . . .
From these, it immediately followed tha t
(
\/-1 .2wx7r W-1 .2wx7r) 2
(
x
x2
..
ex ={l+
.
+ . . .}x{1+1+1
1
+
1 .2
.2+ .}
Therefore
+ ( \/-1 .2w7r + 1)x + (\/-1 .2w7r + 1) 2 x2
1
1 .2
.2w7r
(-/-1
+
x3
+
1)3
+
1 .2. 3
Letting x = 2m =JI (where m and n are both integers) resulted in th e
expansion
ex
=1
2m~r
e 2,17r-,/ -'--L-T
=
1
+
(2wlr - ~/-11 -V-1 .2m7r
J
2n7r - -V-1
1
+ / 2w7r - N/=1 \ 2 (~/-1 .2m7r)2
+
1.2
2n7r - ~/ 1
166
A . Rice
MfM 46 : 2
So, when w = n ,
2mr
e 2n,r- 3-1
1+
A/-1 .2m'Tr
1
cos(2rn7r) +
+ (\/-1.2m7r)2 + (\/-1 .2m7r)3 + . . .
1 .2
1.2 .3
sin(2m'T)
1.
This appeared to satisfy De Morgan, who declared "I see no further objection to Mr . Graves's system, but quite the contrary . . . the extension
may be highly useful, and I am happy to bear testimony to the ingenuit y
which suggested it, and the talent with which it has been carried out" .1 6
[De Morgan 1836b, 253] This left Peacock to be convinced, a feat which wa s
achieved in 1837 by a young Scottish mathematician Duncan Farquharso n
Gregory (1813-1844) .17 In the first edition of his Treatise, Peacock had asserted the existence of logarithmic values common to positive and negative
numbers [Peacock 1830, 569-70] ; but in a paper published in 1837, afte r
providing a further proof of Graves' formula, the young Gregory went on t o
prove the fallacy in Peacock's argument . [Gregory 1837, 134-5] Peacock' s
second edition of 1845 duly contained an acknowledgement of his mistake
and another proof of the Graves result . [Peacock 1842-5, 444-5 ]
Given this eventual acceptance of Graves' generalisation, it is not unnatural at this point to ask why it is rarely (if ever) used . After all, modern
texts on complex function theory employ only the Eulerian definition of th e
logarithm of a complex number, and make no reference to any extende d
formulation . The reason is simple : not only was the generalised version un necessarily complicated, but it was also superfluous the standard Eulerian
logarithmic formulæ are perfectly sufficient to solve exponential equation s
in complex numbers . In short, no real advantage could be gained from usin g
Graves' system in preference to that of Euler .
Nevertheless, the extension of the theory of imaginary logarithms gives an
indication of the renewed interest in Britain at this time in complex numbers, whose existence and legitimacy were now no longer in question . But
this interest extended further ; indeed "from the 1820s through the 1840 s
16 Graves and De Morgan were later colleagues at University College, when Grave s
served as professor of jurisprudence between 1839 and 1843 .
17A great-great-grandson of the great 17th-century mathematician James Gregory
(1638-1675) .
MfM 46:2
167
the study of the foundations of algebra became almost a British monopoly" .
[Hankins 1980, 249] In this respect, as we shall see, the work of both Warre n
and Graves was significant not merely in its own right as innovative mathematical research, but also as a catalyst which would eventually lead to on e
of the most significant mathematical discoveries of the 19th century.
7 "New inexplicables" : the search for triple s
The inspiration (at least in part) behind Graves' work on logarithms ha d
been the idea that there might be other imaginary numbers . Although
Hamilton was sceptical that new imaginaries could be found in that area ,
his interest in complex numbers was aroused by reading Warren's Treatise in 1829 (at Graves' instigation) . [Hankins 1980, 262] Together with th e
prompting of Graves, it was this work which suggested to him the possibilit y
of extending the representation of complex numbers from a plane to three dimensional space . As he later recalled, "To suggestions from that Treatise I
gladly acknowledge myself to have been indebted" . [Hamilton 1853, 31] Thu s
began the search for the new algebraic entity of "hypercomplex" number s
capable of three-dimensional geometrical representation .
Although involved in the same area of research as those so far mentione d
(Peacock, Graves, De Morgan, Gregory), Hamilton's philosophy of the subject was entirely different . 18 As he said to Graves in 1835, "we belong t o
opposite poles in Algebra ; since you, like Peacock, seem to consider Alge bra as a `System of Signs and of their combinations,' somewhat analogou s
to syllogisms expressed in letters ; while I am never satisfied unless I thin k
that I can look beyond or through the signs to the things signifie d" . [Graves
1882-9, II, 143] For an opponent of Peacock's formal approach such a s
Hamilton, algebra was too significant to be reduced to the mere manipulation of arbitrary symbols . Thus, while agreeing that complex number s
were mathematically sound, he maintained that they had yet to be give n
meaning.
18 Because of his shifting attitudes towards algebra, it has been argued that De Morgan' s
views on the foundations of the subject both differed from [Hankins 1980, 249-50], an d
moderately agreed with [Pycior 1983, 223-4 ; Nagel 1935, 4611 those of Hamilton . Similarly,
it has been shown that Hamilton's initial opposition to symbolical algebra softened in late r
years . [Koppelmann 1971, 226 ; Hankins 1980, 310-11]
168
A . Rice
MfM 46:2
Of course, for Peacock and his followers, one could be sure that the result s
obtained in symbolical algebra were both true and meaningful . If they ha d
been derived by means of a consistent set of rules of combination and operation, then they could be viewed as correct within that system . However ,
in order that the results obtained were meaningful, Peacock imposed th e
arbitrary restriction that, the operations of symbolical algebra must obe y
the same fundamental rules as those of arithmetic, including what are no w
known as the commutative and distributive laws . 19 He called this criterion
the "Principle of the Permanence of Equivalent Forms" . 20 [Peacock 1830 ,
104, 1842-5, II, 59 ]
Hamilton's priorities were completely different . Whereas Peacock playe d
down the meaning of the individual symbols, emphasising the rules unde r
which they operated, Hamilton aspired to give meaning to the algebrai c
objects to be sure of obtaining interpretable results . Thus, his view of alge bra was " a science of meaningful symbols governed by necessary principle s
stemming from intuition" . [Pycior 1981, 41] In this approach, he showed th e
influence of the philosophy of Immanuel Kant, who, in his Critique of Pure
Reason (1781), had stated that the intuitive notion of time served as th e
basis for both arithmetic and algebra . From this primitive intuition, Hamilton sought to define the concept of real numbers, and hence the foundation s
of a meaningful algebra .
He did this in a lengthy paper (written in sections between 1833 and 1835 ,
and published in 1837) entitled the "Theory of conjugate functions, or algebraic couples ; with a preliminary and elementary essay on algebra as the
science of pure time" . By basing number on the notion of steps in tim e
from an arbitrary point of origin, Hamilton was able to easily define bot h
positive and negative numbers, since a negative number value would simpl y
19 The terms commutative and distributive were introduced by François-Joseph Servoi s
(1767-1847) in 1814. The laws were used by Peacock in his Treatise, but it was Gregory
who was the first to mention them by name in an English work . [Gregory 1838, 211 ]
The associative law, although implicitly present in preceding works, was first stated b y
Hamilton in 1843 . [Hamilton 1967, 114]
20 Peacock's statement of this principle varied slightly in the two versions of his Treatise.
The 1830 version reads as follows : "Whatever form is Algebraically equivalent to another ,
when expressed in general symbols, must be true, whatever those symbols denote . Whatever equivalent form is discoverable in arithmetical Algebra considered as the science o f
suggestion, when the symbols are general in their form, though specific in their value, wil l
continue to be an equivalent form when the symbols are general in their nature as well a s
in their form ." [Peacock 1830, 104]
MfM 46 :2
16 9
represent a time-step in the opposite direction to that of a positive number .
However, that still left the difficult problem of giving a meaningful definitio n
of complex numbers .
Hamilton solved this by defining number couples (a, b), where a and b were
two real numbers, 21 or moments in time . Defining operations such as
(a,b)+(x,y) =
(a, b) .(x, y) =
(a, b)
(x , y)
(a+x,b+y) ,
(ax - by, ay + bx) ,
ax + by bx - ay 1
C
x2 + y 2 ' x2 + y 2
'
he was able to show that, not only do they give consistent results, but that
any real couple (a, b) is exactly equivalent to the complex number a+b \/-1
in its mode of operation . Thus, by substituting these ordered couples of
(meaningful) real numbers for complex expressions, the whole question o f
giving meaning to imaginary quantities was made redundant . But by th e
time Hamilton published his theory of number couples, the justificatio n
of imaginaries was no longer of importance ; indeed "it was not clear that
Hamilton had created anything more than a new representation of what wa s
already known" . [Hankins 1980, 275] British mathematicians now not only
accepted complex numbers : they wanted to find more !
The endeavour now underway in Britain was to discover if there were num ber triples (a, b, c) which would do for three-dimensional geometry wha t
Hamilton's couples and the standard complex numbers could do for th e
two-dimensional case . Clearly, such an extension of the theory of comple x
numbers would be a far more general and, most importantly, applicable algebraic subject . Yet, as De Morgan recognised, "an extension to geometry
of three dimensions is not practicable until we can assign two symbols, f
and w, such that a + K2 + cc.~ = a l + b 1 S2 + cl w gives a = a l , b = b1 and
c = cl : and no definite symbol of ordinary algebra will fulfil this condition ."
[De Morgan 1839, 177]
Once again, Hamilton's interest in triples was periodically maintaine d
throughout the 1830s and 1840s by John Graves, who himself made sev21 Although Gauss appears to have had the notion of couples as early as 1831, there i
s
no evidence to suggest that Hamilton was aware of this . He seems to have obtained the
idea from Cauchy's Cours d'analyse (1821), which had included the statement that "every
imaginary equation is merely the symbolic representation of two equations between rea l
quantities" . [Cauchy 1882-1938, vol .2, 3]
170
A . Rice
MfM 46 : 2
eral investigations into representing lines in space by means of triples, an d
attempting to multiply them together . One of these systems, which Grave s
developed in 1836, was independently discovered ten years later by his
younger brother Charles . 22 This gave a satisfactory geometrical interpretation of numbers of the form a + bi + cj (i j, i 2 = j 2 = -1), but while
most of the usual laws of algebra held, multiplication was not distributive .
[Hamilton 1853, 37-9] Hence, for formalists such as the Graves brothers ,
this system was not regarded as satisfactory .
Meanwhile, De Morgan was also developing his own theories regardin g
triples and their geometrical representation . In a paper read to the Cambridge Philosophical Society in 1839, he hinted at a use for the cube roots o f
unity as geometrical operators : " in passing from x to -x by two operations ,
we make use in ordinary algebra of one particular solution of 02 x = -x ,
namely Øx = \/-1 .x. An extension to three dimensions would require a
solution of the equation 0 3 x = -x, containing an arbitrary constant, an d
leading to a function of triple value, totally unknown at present ." [De Morgan 1839, 177]
Although this theory was never fully developed, one of the by-products of D e
Morgan's paper was that it prompted Hamilton to re-embark on the ques t
for a sound system of triples . [Hamilton 1853, 41] Since the early 1830s, he
had formulated various triple systems, which for various reasons had failed
to produce unequivocal results regarding multiplication . In particular, they
failed to obey the rule that the product of two non-zero numbers must
also be non-zero - known as the product law . His latest attempt was t o
prove more fruitful, although, since there already exist a number of excellen t
accounts of the eventual outcome of this research, [e .g. van der Waerde n
1985, 179-83 ; Hankins 1980, 293-300] it is only necessary to give a basic
outline of the main details . The first problem came from multiplying tw o
triples together . Hamilton obtained
=
(a + bi + ej )(x + yi + zj )
(ax - by - cz) + (ay + bx)i + (az + cx) j + (bz + cy)ij .
This was obviously not a triple, unless the ij-term could somehow be ac counted for . Also, the norm of a +bi +cj (obtained by multiplying a+bi+cj
22 Charles Graves (1812-1899), professor of mathematics at the University of Dublin ,
and later Bishop of Limerick .
MfM 46 :2
17 1
by a - bi - cj) was expected to be a2 + b2 + c2 since, by analogy to ordinary complex numbers, multiplying complex conjugates a + bi and a - bi
gave a2 + b2 . But in fact it was a 2 + b2 + c2 - 2ijbc. Attempts involving
ij = 1, ij = -1 and ij = 0 all failed to produce satisfactory results, un til he let ij = -ji and defined this as a new unknown : "I made therefore
ij = k, ji = -k, reserving to myself to inquire whether k was = 0 or not" .
[Hamilton 1967, 107 ]
Then, on 16 October 1843, he had a sudden idea of what this k could be:
"there dawned on me the notion that we must admit, in some sense, a
fourth dimension of space for the purpose of calculating with triplets ; or,
transferring the paradox to algebra, must admit a third distinct imaginar y
symbol k, not to be confounded with either i or j, but equal to the produc t
of the first as multiplier, the second as multiplicand, and therefore I was le d
to introduce quaternions such as a + ib + jc + kd, or (a, b, c, d) ." [Hamilto n
1967, 108] By defining k to be a third imaginary quantity, so that k2 = -1 ,
ik = -ki and jk = -kj, he was now able to quickly deduce the fundamenta l
formulae of his new algebra :
2 2=
j2 = k2 =-1,
ij=-ji =k,
jk = -kj =i,
ki=-ik =j.
It thus immediately followed that the norm of a quaternion was
(a +bi
+ej +dk)(a -bi -ej - dk)
a2+b2 +c2+ d2 ( -k2)
- bd(ik + ki) - cd(jk + kj)
a2 + b 2 + c 2 + d2 ,
and that the product of any two quaternions (a + bi + cj + dk) and (w +
xi + yj + zk) would also yield another quaternion :
(aw - bx - cy - dz) + (ax + bw + cz - dy) i
+ (ay - bz + cw + dx) j + (az + by - cx + dw)k .
Hamilton's quaternions had arisen from his search for a coherent syste m
of number triples capable of representing lines in three-dimensional space .
The unexpected inclusion of a fourth term, while ensuring a consistent alge bra, made the geometrical interpretation of quaternions somewhat harder.
While the imaginary (or "vector") parts i, j and k could be said to represent
172
A . Rice
MfM 46 :2
three perpendicular lines, Hamilton interpreted the real part as represent ing " extra-spacial, or simply SCALAR direction", 23 [Hamilton 1967, 359] in
other words, direction on a uni-dimensional line, or a concept independent
of space, such as time .
The "discovery" of quaternions came as a surprise to Hamilton, but more s o
to his British contemporaries . This was for two main reasons . Firstly, sinc e
the products of two distinct imaginaries depended on the order in which
they were multiplied, the commutative law for multiplication was violated .
This meant that he had created the first fully consistent generalised numbe r
system in which the laws of arithmetic no longer held in their entirety .
Thus Peacock's Principle of the Permanence of Equivalent Forms had been
proved to be no obstacle to the formation of a meaningful system of symboli c
algebra.
The second reason concerned Hamilton's "creation" of new algebraic objects; this was the first time that mathematical entities had been deliberatel y
constructed, as opposed to being discovered or evolving from pre-existin g
concepts . As John Graves said : "There is still something in the system which
gravels me . I have not yet any clear view as to the extent to which we ar e
at liberty arbitrarily to create imaginaries, and to endow them with super natural properties ." [Graves 1882-9, II, 443] De Morgan was less surprise d
than Graves at the origin of new imaginary quantities ; indeed, in a paper
completed just nine days before Hamilton's discovery, he had expressed th e
opinion that "new inexplicables might, and perhaps would arise" befor e
long. [De Morgan 1843, 142] Yet even he was taken aback at the idea o f
"imagining imaginaries" . [Graves 1882-9, II, 475 ]
Hamilton's work on complex numbers had initially begun as an attemp t
to place them on a sound intuitive foundation but had quickly develope d
into a search for higher order concepts . It eventually resulted in the discovery/creation of a new and surprising species of number : one that was mad e
up of four distinct components and one whose laws of combination did no t
permit one of the most basic ideas in elementary algebra . It is somewha t
ironic in fact that for a mathematician whose chief priority was to base th e
23 Notice the first use of the word scalar. Although Hamilton also coined the term vecto
r
(in the modern sense) to denote the directed line segment represented by the imaginary
quaternion component, the phrase " radius vector" had been in use for over a century, an d
was still employed at this time [e.g . Warren 1828, 137-8 ; De Morgan . 1839, 186] .
MfM 46 :2
173
concept of number on intuition, Hamilton was responsible for one of th e
most counter-intuitive algebraic creations in the history of mathematics .
8
After
quaternions
The discovery of quaternions heralded a new era in the history of complex
and imaginary numbers . Not only were they now widely accepted by Britis h
mathematicians, but it was also realised that further imaginaries were possible, and that meaningful and consistent systems of hypercomplex number s
were entirely conceivable within the framework of symbolic algebra . The
realisation that mathematicians were at liberty to create new algebras b y
inventing algebraic elements and defining legitimate operations upon the m
was not lost on those who had previously adhered to Peacock's permanenc e
principle .
Among the most active of these were John and Charles Graves and Augustus De Morgan . Although for Hamilton, the question of three-dimensiona l
geometrical representation had been answered by quaternions, these three
continued to try to establish a workable system of triples by exploiting th e
new freedom . In a paper written in October 1844, one year after Hamilton' s
discovery, De Morgan gave his most extensive treatment to date of the subject by constructing five distinct systems of triples . While distinguishin g
his triple systems from Hamilton's quaternion one, 24 De Morgan credited
Hamilton's work "for the idea of inventing a distinct system of unit-symbols ,
and investigating or assigning relations which define their mode of actio n
on each other" . [De Morgan 1844, 241 ]
Unlike Hamilton, De Morgan was not dissuaded by a lack of symmetry i n
his algebras . He wrote : "Sir William Hamilton seems to have passed ove r
triple Algebra altogether on the supposition that the modulus, 25 if any, of
a +br7+cÇ [where 2 = , 172 = (2 =
must be \/a 2 + b2 + c2 . . . but it is
24 In an addendum, De Morgan added that "having since I read this paper in proof ,
examined Sir W . Hamilton's system of quaternions, I may state that, in my view of th e
subject, it is not quadruple, but triple, since every symbol is explicable by a line drawn i n
space" . [De Morgan 1844, 254 ]
25 i . e. the square root of the norm
. The "law of the moduli" holds if the norm of th e
product equals the product of the norms, i .e . (ai + a2 +
+ a 2,) (bi + bz +
+ bet ) _
+ cn . In 1898, Adolf Hurwitz (1859-1919) proved that this law could only
ci + ez +
hold for n = 1, 2, 4 or 8 . [van der Waerden 1985, 185]
174
MfM 46:2
A . Rice
by no means requisite that the modulus should be a symmetrical function o f
a, b, and c . " [De Morgan 1844, 242] He then went on to illustrate this belie f
with a series of various triple systems, all of which had extremely irregula r
moduli. But this drawback was not the only "imperfection" in his systems .
Commenting on a case in which P = a + bra -I- cC, where 71 2 = /-2 = = - 1
he remarked "I did not at first see that though this will give PP' = P'P, it
will not give P"(P'P) = (P"P')P, except in particular cases" . [De Morgan
1844, 249]
Yet while the abandonment of symmetrical moduli and the associative la w
of multiplication in De Morgan's systems was apparently no more imperfec t
than Hamilton's disregard for the commutative law in his, the former remained fundamentally flawed due to their failure to adhere to the produc t
law, a defect which was not present in the latter. The same applied to the
various attempts by the Graves brothers, who constructed similar system s
in the months following the birth of quaternions [e.g . Graves 1847] .
But the search for new complex systems was not limited to triples . In fact, i t
was in a higher dimension that the first successful complex number syste m
after quaternions was discovered . As early as December 1843, John Grave s
constructed a consistent algebra of complex numbers with eight components
which obeyed the law of the moduli . [Young 1848, 338-41 ; Hamilton 1967 ,
648-56] This system of octaves or octonions consisted of the real element
1 , and seven imaginaries, i1 i 22, 23, 24, 25, 26 i i 7 , obeying the following rules o f
multiplication :
2-21
21
2_ 2_ 26
27 _
24 - 25 - i - 2 --1 ,
-i 3 2 2 = -i 5 2 4 = -2 6 2 7 ,
= 2223 = i 4 i 5 = 272 6
=
2_ 2
i2 - 23
22 = 2321 = 2426 = 252 7
-2123
i 3 = 2 1 22 = 24 2 7 = 2 6 25
- 1 2 2 1 = -2 7 24 = -2 5 2 6 1
i4 =
2521
= 2622 = 272 3
- 1 1 2 5 = -2226 = -2 3 2 7 ,
i5 =
2124
= 2722 = 232 6
- 2 4 2 1 = -2227 = -2 6 23 ,
26 =
2224
=
- i4 2 2 =
2127
= 252 3
27 = 2621 = 2225 =
2324
- î1i6
= -2 6 2 4 = -2 7 2 5 ,
-2721
= -2 3 2 5 ,
= -i5î2 = -i4i3 .
This system was also independently created in one of the earliest papers by
the young Cambridge mathematician Arthur Cayley (1821-1895) . [Cayley
1845] In a subsequent article, published two years later, Cayley observed
MfM 46:2
17 5
that " if i s iß, iy be any three of the seven [imaginary] quantities which d o
not form a triplet, then (i niß) .i y = -ia .(ißiy ) . Thus, for instance, (i3i4) .i5 =
-i7 .i5 = -i2 ; but i3.(i4i5) = i3 .i1 = i2 . . . " . Graves ' and Cayley ' s octaves 2 6
were therefore neither commutative nor associative, "which is still a wide r
departure from the laws of ordinary algebra than that which is presente d
by Sir W. Hamilton's quaternions" . [Cayley 1847, 257-8 ]
From this point, a variety of new systems of hypernumbers began to emerge .
Hamilton created a new system of biquaternions (quaternions with complex
coefficients) in 1844, although he noticed that they failed to obey the product law . [Hamilton 1853, 650] A generation later, another young Cambridge trained mathematician, William Kingdon Clifford (1845-1879), produce d
another form of number, which he also termed biquaternions, although hi s
definition was very different from that of Hamilton . 27 [Clifford 1873] These
satisfied the product law but not the associative . The introduction of "Clifford algebras" five years later gave a generalisation for systems with 2 n
elements ,î 2n_ 1 , where i 2a = -1 and i a i b = -i b i a for a b.
[Clifford 1878] But Clifford, although British, had been more influence d
by recent German work on hypernumbers, particularly that of Herman n
Günther Grassmann (1809-1877), than by his contemporary countrymen .
This provides an indication that, by the middle of the century, researc h
into higher-order complex numbers was no longer dominated by Britis h
mathematicians . Indeed it was to be elsewhere, especially in Germany, that
the major advances in this area were achieved in the second half of th e
19th century. This work culminated in Adolf Hurwitz's 1898 proof that th e
only linear associative algebras which satisfy the product law are the rea l
numbers, complex numbers, real quaternions and Clifford's biquaternions .
[Kline 1972, 793] Thus it was finally confirmed that a triple system of th e
kind originally envisaged by Hamilton, De Morgan and the Graves brother s
was impossible .
While the latter half of the 19th century witnessed profound development s
on the algebraic implications of quaternions, similar extensions were als o
being made to the geometrical side of the subject . Through the work of
Scottish mathematicians Peter Guthrie Tait (1831-1901) and James Cler k
26 Because Cayley was the first to publish his findings, octaves are known today a
s
Cayley numbers .
27 Clifford's biquaternions are of the form p + wq, where p and q are real quaternions, w
commutes with every real quaternion, and w 2 ' = 0 or w 2 = 1 .
176
A . Rice
MfM 46:2
Maxwell (1831-1879), the three-dimensional properties of quaternions were
adapted for use in physics . These ideas were developed by Oliver Heavisid e
(1850-1925) in England and Josiah Willard Gibbs (1839-1903) in the Unite d
States to found the subject of vector analysis, one of the most valuabl e
tools in modern applied mathematics . [Crowe 1967] Thus, by the end o f
the century an algebraic system derived from imaginary numbers would b e
employed to model the real world .
In the space of one hundred years, complex numbers had gone from bein g
concepts of uncertain meaning and dubious mathematical validity to univer sally recognised mathematical entities of acknowledged importance . As the
basis of far-reaching extensions to both algebra and geometry, they acted a s
a catalyst for the vast developments in those fields in the latter part of th e
19th century. But as far as British mathematicians of the mid-19th century
were concerned, all this lay ahead . While the future of complex numbers in
mathematics was secure, the outlook was still uncertain . But to those who
advocated the more risk-averse approach which had characterised much o f
the British mathematics of the late-18th and early-19th centuries, Augustus De Morgan had the following words [Richards 1987, 28] : "The mott o
which I should adopt against a course which seems to me calculated to sto p
the progress of discovery would be contained in a word and a symbol remember -/-1 ."
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MfM 46:2
18 1
Bellavitis's Equipollences Calculus and his Theory
of Complex Numbers
Paolo Fregugli a
Abstract
The aim of this contribution is to present a detailed analysis o f
Bellavitis's work. In particular, we will examine Bellavitis's theor y
about the geometrical nature of complex numbers and the connection s
with his equipollences calculus . Afterwards, we will examine some as pects of the resulting theory of algebraic equations .
Introduction
Although Bellavitis's analysis of the theory of complex numbers was of secondary importance compared to what other mathematicians had done a
little before him on this subject, it was the geometrical representation of th e
complex numbers that led him to make the equipollences calculus . More over, during the first half of the XIX century, in Italy it was he who took
most interest in the "nature" of complex numbers . The aim of the present
contribution is to go through some essential features of the equipollence s
calculus and to look at some interesting links between this calculus an d
the theory of complex numbers . Synthetically, we can say that Bellavitis' s
goal was to carry out Lazare Carnot's program regarding the position calculus and hence to establish a parallelism between algebraic calculus an d
geometrical calculus in the wake of the barycentric calculus .
Giusto Bellavitis was born in 1803 at Bassano del Grappa, near Padua ,
where he studied . He worked as a clerk at the municipality of Bassano ,
* Dipartimento di Matematica pura ed applicata, Università di L'Aquila, Domu s
Galilaeana, Via S .Maria 26, I-56126 Pisa, Italy .
182
P. Freguglia
MfM 46: 2
until 1843, when he was called to teach at the Vicenza "liceo" . Throughou t
this period he maintained close contact with the mathematical circle at th e
University of Padua . In 1845 Bellavitis became a full professor of descriptiv e
geometry at Padua University. His colleagues included such interesting Italian mathematicians as Conti, Minich and Turazza . Moreover, Bellavitis late r
became rector and a senator. When he died in 1883 he left approximately tw o
hundred works mainly on geometry, in particular the geometrical calculus
(equipollences calculus) and its applications and on the theory of comple x
numbers. l
1 The sources which inspired Bellaviti s
Bellavitis's intellectual itinerary was rather singular, but at the same tim e
emblematic of a certain cultural atmosphere present in Italy during the firs t
half of the XIX century. As a starting point for his research he took theme s
connected with Lazare Carnot's Géométrie de Position 2 (1803) (analogous ,
for instance, to J .F .Français 3 ) . However, he was also interested in the prob lems which were being discussed between the end of the XVIII and the first
half of the XIX century regarding the geometrical representation of comple x
numbers . In 1847 he wrote : 4
The geometrical representation of the imaginary numbers, which ha d
been almost completely forgotten when I drew the equipollence s
method from it some years ago, has now become the object of several essays in the Philosophical Magazine.
However, from a theoretical point of view, the equipollences calculus come s
first, as this method justifies the theory of complex numbers . So we will
begin with the equipollences calculus . In 18765 Bellavitis reviewed the work s
' Regarding the analysis of Bellavitis's work we have already devoted some papers (se e
P.Freguglia [1991], [1994] and with G .Canepa [1991]) and a book (see P.Freguglia [1992]) .
2 The word "equipollence" is drawn from the Géométrie de Position, but with a different
meaning, see p .83 of L.Carnot [1803] .
3 See J.F.Français [1813] .
4 See "Saggio sull'algebra degli immaginari", G .Bellavitis [1847], p .249, "La rappresentazione geometrica degli immaginari, the era quasi affatto dimenticata quando alcun i
anni or sono io ne traevo il metodo delle equipollenze, forma ora oggetto di frequentissim e
comunicazioni nel Philosophical Magazine" .
5 See G.Bellavitis [1876] pp . 469-473 .
MfM 46 :2
Bellavitis's Equipollences Calculus
18 3
of those mathematicians (Buée, Argand, Français, Mourey, Warren) who ,
before him, had analysed the "nature" of complex numbers . The author
whom Bellavitis considered to have influenced him most was Buée 6 , who in
an essay published in Philosophical Transactions in 1806 had suggested a
geometrical representation of the imaginary numbers which was afterward s
used as a basis for Bellavitis's approach . Buée's fundamental idea is wel l
summarised in this short passage drawn from his essay : 7
So \/-1 is the sign of perpendicularity, whose characteristic is tha t
every point belonging to the perpendicular is equally far from [two ]
points placed at the same distance from either side of the foot (of th e
perpendicular) . The sign \/- 1 expresses all this and it is the only sig n
that does this.
It seems that for Buée the nature of \/-1 (imaginary unit) is purely geometrical and it has no other meaning.
As a matter of fact the main concern for mathematicians such as Buée ,
Français, Argand, etc . was to find the "true" nature, the "true" essence o f
the imaginary numbers . Argand, 8 for instance, saw the imaginary unit as a
geometrical mean between the real positive unit and the real negative unit ,
that is :
-1 : -V-1 = \/-1 : 1 or
-1 :
- -V-1 = \/- 1 :
1.
But we must not forget that the "true" theory of ratios was traditionally a
geometrical theory.
The accepted view gave an epistemologically acknowledged foundation onl y
to the positive real numbers, which were connected to Euclidean geometr y
6 See A . Buée [1806] . Bellavitis says : "In reality, it was while I was studying a geometrical representation of imaginary numbers, which Buée had proposed, that I had th e
first idea (1832) about the equipollences method [ . . . ]" ("Fu veramente considerando una
rappresentazione geometrica degli immaginarii proposta da Buée the a me venne (1832 )
la prima idea del metodo delle equipollenze [ . . . ]"), see G .Bellavitis [1876], p .58 . See als o
Schubring's paper in this volume .
' See A . Buée [1806] p .23, "Ainsi \/-1 est le signe de perpendicularité, dont la propriét é
caractéristique est, que tous les points de la perpendiculaire sont également éloignés de
points placés à égales distances, de part et d'autre de son pié . Le signe -/-1 exprime tout
cela, et il est le seul qui l'exprime" .
8 See J .R.Argand [1806] p .6 .
184
P . Freguglia
MfM 46 : 2
(theory of quantities) . Because of this none of the negative roots of algebraic equations were accepted because they disagreed with the synthetic Euclidean geometrical interpretation . In this respect, it is interesting to make
a comparison with D'Alembert's article "Negatif" in the Encyclopédie and
with his essay "Mémoire sur les quantités négatives", which is in the VII I
book of his Opuscules mathématiques . These papers were well known to
Lazare Carnot and also, indirectly, to Bellavitis . Now we turn to examin e
the equipollences method .
2 The equipollences calculu s
In his principal work, written in 1854, Bellavitis motivated the theory a s
follows9 :
This method [of the equipollences] satisfies Carnot's wish to find an
algorithm which represents at the same time the length and the position of the various parts of a figure ; we therefore have, directly, elegant easy graphic solutions for geometrical problems . The equipollences method includes as particular cases the method of parallel o r
polar co-ordinates, the barycentric calculus etc . [ . . . ] . He who knows
the principles of Carnot's Géométrie de Position, will easily follow m e
[ . . . ].
The equipollences calculus is a geometrical algebraic structure analogou s
to that of a vector space where the multiplication and division operation s
(between vectors) are defined . Bellavitis never considers the possibility o f
an axiomatic theory. His position is strongly heuristic . He starts from th e
definition of parallelism (23rd definition [opo(] of the I st book of Euclid's Elements) and from the following principle (which in this theoretical attitud e
is not a theorem) :
[P .1] : "two angles which have parallel sides are equal" (see Figure 1 )
9See G.Bellavitis [1854] p .2, "Questo metodo [delle equipollenze] soddisfa un desiderio
del Carnot di trovare un algoritmo the rappresenti nello stesso tempo e la grandezza e la
posizione delle varie parti della figura; ne risultano quindi per via diretta eleganti e semplic i
soluzioni grafiche dei problemi geometrici. Il metodo delle equipollenze comprende come
casi particolari i metodi delle coordinate parallele o polari, il calcolo baricentrico ecc . [
] . Chi sia abituato ai principi della Géométrie de Position di Carnot troverà facile
seguirmi [ . . . ]" .
MfM 46 :2
18 5
Of course this principle is equivalent to Euclid's proposition (theorem) I,29 :
if r s and t H t' then a = b .'° Thus for Bellavitis two oriented segments AB
and CD are called equipollent if they have the same length and the same
direction . We will denote this fact by writing as follows :
AB
= CD .
Instead of calling AB an "oriented segment", which is the term used by
Bellavitis, we will often call AB a "vector applied in a point", which is a
more modern mathematical term . Now in the plane, if a horizontal straigh t
line is given, Bellavitis defined for an oriented segment AB the inclinatio n
operator as follows . Consider a point O and a fixed straight line OR; then
the inclination operator can be defined by means of the relations (see Fig .
2) :
D'
OA'
R
Figure 2:
inc AB = ang B'A'R which means the angle B'A'R oriented in an anticlockwise direction. Here A'B' = AB and A' - O . It follows that inc OR =
10 We must observe that Euclid's I,29 is logically equivalent to Euclid's Vth postulate .
186
P. Freguglia
Mf'M 46 : 2
0 and ang BAD' = inc AB - inc CD . Bellavitis gives the other followin g
fundamental principles :
[P.2] inc BA = inc AB ± 180 °
[P.3] (the rule of parallelogram) AB H- BC = A C
[P.4] AB + 0 r- 0+ AB = AB (existence of vector 0 which has length equal
zero and no direction )
[P.5] AB + BA = 0 or AB = -BA .
From AB + BC AC we have AB + BC + CA = 0 or AB = AC - BC .
Bellavitis also implicitly considers the composition + to be associative an d
commutative .
He gives this following definition :
AB tCD, where t e R, if AB and CD have the same direction . Howeve r
the length and the orientation of tCD depend on t.
Moreover Bellavitis defines :
i) AB = BD • EF if gr AB = (gr CD) • (gr EF) and inc AB = inc BD +
inc EF, where gr AB means length of AB .
ij) BD ^ AB/EF if gr BD = (gr AB)/(gr EF) and inc BD = inc AB inc EF.
With regard to the orientation of AB, every ambiguity is solved by takin g
each oriented segment to its respective equipollent which has its origin i n
the point O of the fixed straight line . The operator inc - as we can easil y
observe - has the same formal behaviour as the operator log . Moreover ,
as is well known, the equipollences calculus is a geometrical plane calculu s
which presents some difficulties if we want to expand it to three dimensions .
Bellavitis gave some canons which are helpful for facilitating the calculus .
These canons (twelve in all) are in reality similar to lemmas (preliminary
theorems) . The fifth canon deals with the conjugate A'B' of an oriente d
segment AB (A'B' = cj AB) that is its mirror image in OR (see Fig .3) .
Clearly gr(cj AB) = gr AB = gr A'B', inc(cj AB) = - inc AB = inc A'B' .
cj is like a 180 degrees overturning operator which carries the semiplan e
ABC on to the semiplane A'B'C' . The notion of a conjugate of an oriente d
segment is, of course, suggested by the analogous notion relating to comple x
numbers . Bellavitis also examines the behaviour of the operator y whic h
turns an oriented segment AB by 90 degrees, that is :
yAB = A'B' if and only if AB is perpendicular to A'B' .
MfM 46 :2
18 7
R
Figure 3 :
The author says that -y behaves in calculus as V-1 (which he calls "ramuno", an abbreviation of "radice di meno uno", or in English "root o f
minus one") . Bellavitis states that all the elementary geometrical theorem s
(and also those relating to projective geometry, such as Desargues's theore m
(plane case) about homological triangles) can be proved by the equipollence s
calculus. Bellavitis writes : 11
In elementary geometry the method of equipollences can be use d
[ . . . ] to deduce from a few easily demonstrable principles all th e
theorems which we use to teach, and many others besides [ . . . ] .
To prove Desargues's theorem about homological triangles we need only the :
[Th. 2 .1] Second canon . If the terms of a binomial equipollence have different inclinations, then every term of it is zero .
That is, let us suppose that mIL ^ nMN (binomial equipollence), wher e
IL and MN are different from zero . We know that if inc IL
inc M N
then mIL and nMN are not equipollent . But we suppose mIL nMN ,
therefore IL = MN O .
The statement, which Bellavitis gives, regarding Desargues's theorem o n
homological triangles is the following : "If the vertexes of two triangles AB C
11 See G .Bellavitis [1843] p .5, "Nella geometria elementare il metodo delle equipollenz e
pue) servire [ . . . ] a dedurre da pochissimi principi di facile dimostrazione tutti i teorem i
che soglionsi insegnare, oltre molti altri [ . . . ]" .
188
MfM 46 : 2
P. Freguglia
and A'B'C' are in a straight line with the fixed point S, then the intersectio n
points T, U, V of their corresponding sides are also in a straight line" 12 (se e
Appendix Fig .10 and ibid . § .A .1 for the proof) . On the other hand, to prove
Pythagoras's theorem (see Appendix ß .A .2) we need 13 the next two canons .
The third canon deals with the following situation :
LM +IVIN +NL
O.
[Th . 2 .2] Third canon . If two terms of a trinomial equipollence hav e
equal inclinations, the third term (if we decide to transpose all three term s
to only one member) will have an inclination which will differ 180 degree s
from that of the other two and its length will be equal to the sum of th e
length of the first two terms .
The proof of this canon is trivial .
L
N
M
Figure 4 :
[Th . 2 .3] Fourth canon . If we compare the terms of a trinomial equipol-
lence with the terms of the identical equipollence :
LM+MN +NLAO.
and we ascertain inc ML + inc LN = 2 inc MN (and the three inclinations
are unequal) then it follows that gr LM = gr NL, and vice versa : if gr LM =
gr NL then inc ML + inc LN = 2 inc MN .
Proof. In Fig . 5 let MLA L'N and L'M + MN + NL'
inc L'N
= ß, inc ML' =
S and inc
MN
If we substitute T = a + ß,
ML =
=2T .
S = a + ß + a l into the hypothesis we have
6+,3
( hyP . ) :
0 and inc
= T . Let it be given that
ß+a+3+al = 2(a+ß) ,
12 See G .Bellavitis [1854] pp . 20-21 .
13 See G .Bellavitis [1854] pp .13 and foil .
MfM 46:2
189
M
Figure 5 :
so that a + a 1 = 2a that is a = ctr l and hence :
(th.) :
gr ML' = gr L'N.
Conversely, assume:
(hyp .) :
gr ML'
= gr L'N
then a = a 1 so that S = 2a + O . By adding ß to the two members of the
last expression we have :
+0=
2a
+2 O
but T = a + ß and hence :
(th.) :
S + /3 = 2T .
q
Epistemologically these demonstrations of theorems from elementary geometry using equipollences calculus offer a certain interest . It is a new way t o
see things and an anticipation of the modern ideas of geometrical algebra ,
where the main concept of vectorial space has a central role . 14 Indeed, wit h
Bellavitis's help we have a calculus which governs all geometry and whic h
enables us to find the solutions to geometrical problems .
" For instance see Emil Artin Geometric Algebra (1957), Interscience Publishers, Inc . ,
USA, John Wiley and Sons, 3rd ed . 1964 .
190
P . Frequglia
MfM 46:2
3 The theory of complex number s
In "Saggio sull'algebra degli immaginari" Bellavitis asserts :''
[ . . . ] the [geometrical] representation of imaginary quantities was of ten used by Analysts long before I deduced my method of equipollence s
from it ; Cauchy also utilises it a lot, but always as a tool to expres s
more clearly certain situations relating to imaginary quantities, bu t
not for their sole definition : instead I am going to take it as the tru e
definition and from it I deduce the properties of imaginary numbers .
Bellavitis clearly believed that the geometrical road was the main one to b e
followed in approaching the foundation of algebra . According to Bellavitis
the algebra of real numbers also has its basis in the geometry of the straigh t
line . Bellavitis considered the algebraic calculus to be a useful representativ e
simplification . Thus by means of the equipollences calculus Bellavitis algebraized the complex numbers . First we must say that a complex numbe r
is represented by an oriented segment in a plane . 16 So addition and sub traction are operations like those between two oriented segments . Also for
multiplication (and division) Bellavitis applies the equipollences calculus ,
but in this case for a better geometrical agreement he gave a geometrica l
construction where the length of the resulting oriented segment is graphically determined . 17 This construction is completely analogous to the on e
given by Argand . 18
'See G .Bellavitis [1847] p .247, "[ . . . ] la rappresentazione [geometrical delle quantità
immaginarie, fu data da parecchi Analisti molto prima the io ne deducessi il mio metod o
delle equipollenze ; anche Cauchy la adopera non rade volte, ma sempre come un mezzo
di esprimere più chiaramente qualche circostanza relativa alle quantità immaginarie, no n
già come l'essenziale ed unica definizione delle medesime : io invece la prenderb come la
vera definizione e da essa dedurrà le proprietà degli immaginari" .
16 Bellavitis gives also the trigonometrical representation of a complex number (se
e
G.Bellavitis [1847] p .260) . He writes :
z/u
= OM = OP + OQ = z cos u + z sin u.V,
where .3 = i and u E Q is the number (whole or fractional) of right angles (7r/2) .
Obviously, ,vu = cosu + \/ sinu, for instance when u = 1 we have: ,‘/1 = cos(7r/2) +
.3 sin(7r/2) = .3.
17 0f course the subtraction and the division are deduced respectively from the additio n
and the multiplication .
18 See J .R.Argand [1806] p .21 .
MfM 46:2
19 1
The powers are iterations of the product . For the cases of the square and o f
the cube Bellavitis gave an explicit geometrical construction .
Y
X
1
Xl
Figure 6:
Square case : where OX 1 ^ -OX; OX2
OY means gr OY = (gr OX) 2 ,
inc OY = 2 inc OX ± 2krr . 19 The triangle OXY is made similar to triangle
01X and it is possible graphically to determine the point Y . We have (fo r
the lengths) :
01 : OX = OX : OY
with 01 = 1
(1)
hence :
gr OY = (gr OX)2
Moreover, if we pose OX 1
-OX, that is gr OX 1
angX1O1 = angX1OY) we have from (1) :
gr OX (and
01 : OX1 = OX 1 : OY.
Hence ~/OY OX, OX 1 or \,/OY ±OX .
As we can see, Bellavitis was concerned to find the lengths of oriented segments (the moduli of complex numbers) by methods of Euclidean syntheti c
geometry . The constructive geometrical features are prominent in Bellavitis's work .
19 The behaviour of values of k is according to De Moivre's formula
.
192
P . Freguglia
MfM 46 :2
Finally we want to discuss what Bellavitis calls the fundamental theorem
(or fundamental canon) for the equipollences calculus. This canon - as w e
will see - plays an essential role in the theory of complex numbers . It is
not difficult to demonstrate this theorem .
[Th . 3.1] . "In the equipollences / . . . J we can do all operations and al l
calculations which would be lawful in the case of equations, and the resultin g
equipollences are always correct ". 20
Besides we have the following corollary :21
[Th. 3.2] . "If any relation among the distances of points of a straight lin e
is found and expressed by an equation, we also have a corresponding relation among the points of a plane which transforms the equation into a n
equipollence " .
For example (see Fig .7), for any position of A, B, C, D and E on a straight
line, where EB = CD, we have :
AE•BC=AB•BD-AC•CD .
(2 )
If we put AE = x, EB = CD = a, BC = y and we substitute it into (2 )
then we have the identity :
x•y=(x+a)(y+a)-(x+y+a)a .
In the plane this corresponds to the following equipollenc e
AE•BC--AB•BD-AC•CD
(3)
(see Fig .7) .
Bellavitis's interpretation of these theorems is that the algebraic propertie s
of the complex numbers (oriented segments in a plane) and those relatin g
to the real numbers (oriented segments in a straight line) are the same .
20 See G .Bellavitis [1876] . The results were already found in 1832 (see G .Bellaviti s
[1832]) .
21 See G .Bellavitis
[1876] p .455, "[ . . . ] trovata tra le distanze dei punti di una retta
una relazione qualunque espressa in equazione, si ha una relazione tra i punti di un piano
mutando 1'equazione in equipollenza" .
MfM 46:2
x
A E
a
a
Y
B
193
C
D
Figure 7:
4 Algebraic equations
In § II (entitled "About algebraic equations") of his essay on the algebr a
of imaginary numbers (1847), Bellavitis examined some applications of hi s
theory of complex numbers to the theory of algebraic equations . His treat ment is based on geometrical constructions, taking the general equation o f
fourth degree as an example . Afterwards he studies the numerical solution s
of equations . But Bellavitis's essential and partially original contributio n
concerns the following . He states : 22
For any value of the unknown X we will build the polynomial Y =
D + CX + BX 2 + AX 3 +X4 by drawing the DC I equal and parallel t o
the straight line [oriented segment] which is expressed by the produc t
CX ; similarly, C1 B 1i B 1 A l , A 1 Y are equal and parallel to those give n
by the other monomials BX 2 ,AX 3 ,X 4 . (The polygon ODC1 B I A I Y
of the figure [see Fig .8] corresponds to the case where X = 1, so that
22 See G .Bellavitis [1847] p .265, "Per un qualunque valore dell'incognita X si costruir à
il polinomio Y = D + CX +BX 2 + AX 3 + X 4 tirando la DC1 eguale e parallela alla rett a
espressa dal prodotto CX ; similmente le C1B1, B1A1, A 1 Y egauali e parallele a quell e
espresse dagli altri termini BX 2 ,AX3 ,X 4 . (I1 poligono ODCi B 1 A 1 Y della figura [see
Fig .8] corrisponde al caso X = 1, sicchè DC 1 = OC, C1B = OB, B1 A,. ~ OA, A 1 Y = 01 ,
esprimendo con il segno = la condizione di due rette di essere uguali, parallele e dirett e
per lo stesso verso) . Al variare di X variano anche i punti Cl , B 1 , A1, Y ; e se X è una
radice dell'equazione il punto Y cade in 0" .
194
P. Freguglia
MfM 46:2
DC1 = OC, C1 B OB, B1 A 1
OA, A l Y = 01 where the sign =
expresses the condition that two straight lines [oriented segments] ar e
equal, parallel and going in the same direction) . When X changes valu e
C l , B1 , A l , Y also change values ; and if X is a root of the equation the n
the point Y coincides with O .
First he says that it is necessary to distinguish a formal algebraic equation ,
for instance
x4
+ ax3 + bx2 + ex + d = 0 [with a, b, c, d, x
E
C]
from the corresponding equipollence . The former is a purely syntactic ex pression, conceived as a combination of symbols which obey the laws o f
algebra (logistica speciosa, according to Viète) . The latter, on the othe r
hand, is an algebraic-geometrical expression, for instance :
X4 +AX 3 --f-BX 2
+CX+D= O
where A, B, C, D, X are oriented segments (vectors) .
Bellavitis ' s treatment of the various theorems is a little obscure . Thus we
will try to reconstruct the essential meaning of Bellavitis's ideas by givin g
them philological attention . We think that Bellavitis, who always had a
great respect for the mathematical tradition, related his construction t o
the theme of the geometrical construction of algebraic equations as it wa s
studied by the algebraists of the XVI century (and the XVII century) an d
differently by Descartes . 23 Substantially, Bellavitis obtained the following
results :
[Th . 4 .1] .
Given an algebraic polynomial function :
y = aoxn + alxn-1 + . . + an,
(4)
23
The geometrical constructions of algebraic equations which we find in the treatment s
of algebrists of the XVI century (Cardano, Ferrari, Tartaglia, Bombelli, Bonasoni, Stevin )
constitute the `synthetic geometrical theory of algebraic equations' (see P .Freguglia "Su r
la théorie des équations algébriques entre le XVI et le XVII siècle", Bollettino di Stori a
delle Scienze Matematiche, vol . XIV(1994) fasc .2, and "Sul principio di omogeneità dimensionale tra Cinquecento e Seicento", Bollettino UMI 1999, (8) 2-B) . These construction s
are different from those made by Descartes in his Géométrie (see H .J .M .Bos "Argument s
on Motivation in the Rise and Decline of a Mathematical Theory ; the `Construction o f
Equation', 1637-ca . 1750", Arch. for Hist . of Exact Sciences, XXX (1984) ) .
MfM 46 :2
A1
19 5
Y
B1
C1
D
C
1
Figure 8:
then an equipollence :
Y - AoX n + A1 X' I + . . . +. An
( 5)
exists and it represents (4) in Il82 (Euclidean plane) .
We will present the construction as regards (4) for the case of fourth degree ,
in the same way Bellavitis did . For any x E C we can give the following
construction (see Fig .8) :
1. X can represent any complex number, but let us consider (wit h
Bellavitis) the case X = 1 to make the drawing easier .
2. We will draw (see Fig .8) the vectors - points 1, A, B, C, D (coefficients of the equation, which are in general complex numbers) .
3. DCI - OC; C1BI - OB; B1A1 OA and then A1Y 01 .
4. So we have the open vectorial polygon ODC 1B1A1Y .
We have gone from O as far as Y . We obtain the same result if we ad d
vectors in succession as follows :
OD+O C
OC1 ,
OC1 + OB
OBI + O A
0A1 +01
OBI ,
OA 1 ,
OY.
196
MfM 46 : 2
P . Freguglia
Therefore :
OY ~ O1 +OA +OB +OC +OD
and also :
OY-- 14 +OA•1 3 +OB-12 +OC -1+OD
and thus, by putting X in place of 1 (of course the variable of Y depends
on X) :
Y-=X 4 +AX3 +BX2 +CX+D ,
which is (5) for the case of fourth degree .
Now, if F(x) = 0 is an algebraic equation, we will associate to the equatio n
F(x) = 0 an identity namely F(xo) = 0, where xo is a root of F(x) = O.
Moreover, an (open or closed) vectorial polygon is a finite succession o f
vectors where the origin of one coincides with the end of the previous one ,
save that we say otherwise . It is possible to have a closed vectorial polygo n
where its first vector has the origin which coincides with the origin of it s
last vector (see for example Fig .9, where OB is the first vector and OA is
the last vector of the closed vectorial polygon) . We have :
[Th. 4 .2] . If Xo is a geometrical representation of a root xo of the equation :
xn +alxn-1 +-• +an=0,
(6)
then a closed vectorial polygon (of n + 1 vectors) exists in such a way tha t
the origin of the first vector coincides with the origin of the last vector an d
they both coincide with the origin O (for example see the closed vectoria l
polygon OBA of Fig.9).
Proof. Indeed if xo is a root of (6) then to equation (6) we associate th e
identity :
xå + a1 4 -1 + + an =
O.
(7)
But (7) is (4) for x = xo and y = 0 . According to the earlier theorem 4 .1
we can then represent (7) in R 2 by the (vectorial) equipollence :
Xp + A1Xå-1 + .
+ An n 0,
(8)
MfM 46:2
19 7
where Xo represents xo in R 2 . Fig .9 represents the special case of th e
quadratic equation .
x2 - 2x + 4 = 0,
with the root xo = 1 +'1
making the angle a = 60° with the fixed axis .
Xô-2Xo+4o 0
Xô -- 2Xo
4
OBA is the closed vectorial polygon of 3 vectors :
Xô - OA ; -4 w OB ; 2Xo = BA .
Figure 9 :
Bellavitis wrote this essay in 1847 . Later, Karl Culmann in his Graphisch e
Statik (1864, 1865) proposed a method of solving algebraic equations b y
using graphical-numerical approximate values . 24
From a foundational point of view Bellavitis's position regarding the fun damental theorem of algebra is in line with Français's formulation of thi s
theorem, that is : 25
All roots of an algebraic equation of whatever degree are real, and
they can be represented by segments whose lengths and directions ar e
given .
24 Culmann was inspired by the works of some of his contemporary mathematician
s
(Stamm, D'Egger, Jäger), see K.Culmann [1880] pp.16-19 .
25See J .F . Français [1813] p .73, "THÉORÈME IV. Toutes les racines d'une équation de
degré quelconque sont réelles, et peuvent être représentées par des droites données de
grandeur et de position" .
198
P. Fregu.glia
MfM 46 : 2
5 Conclusio n
Bellavitis had an extensive correspondence 26 above all with Italian mathematicians (for example Fusinieri, Casari, Genocchi, Chelini, Minich, Piola) .
He also had contacts with foreign (European) mathematicians such as th e
Frenchmen C .A. Laisant and M .J .Hoüel, and the Bohemian Zahradnik, wh o
all translated and elaborated 27 Bellavitis's work, in particular his equipollences theory. Bellavitis was one of the first mathematicians in Italy wh o
knew and studied W .R .Hamilton's work regarding quaternions . He translated the theory of quaternions into his own language and gave a lectur e
on this subject at the Istituto Veneto di Scienze, Lettere ed Arti on March
21st 1858 . In the same year he wrote and published an essay about the relationship between quaternions and the equipollences method . 28 The ideas
of H .Grassmann were also known to and studied by Bellavitis . 29
Lastly we can try to draw some conclusions about Bellavitis's ideas on com plex numbers . We can conclude that Bellavitis's contribution was interest ing, but conceptually it imparted no particularly new ideas compared t o
the contributions of other authors who studied the same theme during th e
same period . The epistemological approach of Bellavitis was closely relate d
to that of Buée and Français . Nevertheless, we must give credit to Bellaviti s
for having developed the structure of the calculus and for having applied i t
to the foundation of the theory of complex numbers and to the deductio n
of theorems of elementary geometry (see the following appendix) and so t o
have proposed a synthetic and constructive method which is not in contrast
with analytical geometry.
Appendi x
A.I . Bellavitis's proof of Desargues's theorem on homological
triangles
(See G .Bellavitis [1854] pp.20-21)-( Theor .) : "If the vertexes of two triangles
ABC and A'B'C' are in a straight line with the fixed point S, then the
26 See Appendix
B (by G .Canepa) of P.Freguglia [1992] and G .Canepa [1994] .
27 See M .J .Hoüel [1869], C .A .Laisant [1874], [1887] .
28 See G .Bellavitis [1858] .
26 See G .Bellavitis [1854b] .
MfM 46 :2
199
intersection points T, U, V of their corresponding sides are also in a straigh t
line "
(Proof) : The given conditions on the triangles ABC and A'B'C' can b e
translated into the following equipollences (see Fig .10) :
SA' aSA,
SB'
bSB, SC '
cSC .
A'
T
S
Figure 10 :
where a, b, c are numerical coefficients . Moreover the condition that V belongs to the straight line AB is expressed by AV = nAB, or if we reduce
to SA, SB and SC, by :
SV
=
SA + AV = SA + nA B
= SA - nBA ^ SA - n(SA - SB) z (1 - n)SA + nSB . (9)
But V must also belong to A'B', hence analogously we will have :
SV
(1 - m)SA' + mSB '
all - m)SA+ bmSB .
Comparing (9) with (10) we have :
(1 - n)SA + nSB
a(l - m)SA + bmSB,
(10 )
200
P. Freguglia
MfM 46 : 2
which by the second canon implies tha t
(1-n)=a(1-m)
and
n=bm .
(11 )
From the system (11) we can deduce the value of m ., and if we replace it i n
(10) we have:
SV = [(1 - b)/(a - b)]SA ' + [(a - 1)/(a - b)]SB'.
(12 )
Analogously we have :
ST = [(1 - c)/(b - c)]SB ' + [(b
1)/(b - c)]SC '
(13)
SU = [(1 - a)/(c - a)]SC ' + [(c - 1)/(c - a)]SA' .
(14)
and
From (13) we find SB' and from (14) we find SA' . If we replace the respective
expressions in (12), we hav e
(a - b - ac + bc)SV + (c - a + ab - bc)SU + (b - c - ac - ab)ST -0. (15)
But since SV = TV + ST and SU -- TU + ST, we get from (15) :
(a - b - ac + bc)TV + (c - a +
ab -
bc)TU = O .
(16 )
From the second canon we conclude that TV has same inclination as TU,
i.e. that the points T, U, V are in the same straight line . Hence Desargues' s
theorem is demonstrated .
A .2. Pythagoras's theorem as proved by Bellavitis
(See G.Bellavitis [1854] pp .13 and foil .)
Fig.11), where AC is its median, the n
AB + BC = AC
that is
Given any triangle ABD (see
AB = AC + CB .
Similarly
AD= AC - CB .
Mf1V1 46: 2
A
20 1
D
Figure 11 :
By multiplying these two equipollences with each other we have :
AB•AD=(AC+CB)(ACCB)=(AC) 2 -(CB) 2
or AB•AD+(CB) 2 -(AC) 2 =AB•AD+(CB) 2 +AC•CA=0 . (17 )
Besides, if we suppose that CB is perpendicular to AC, that is
inc CB =incAC+90°,
(18 )
2 inc CB = 2 inc AC f 180°
(19 )
we also have :
Moreover :
since (CB)2 = CB • CB
we have
2 inc CB = inc CB + inc CB
(20 )
and
since - (AC) 2 = AC . CA it follows that
inc AC H- inc CA = inc AC + inc AC ± 180° = 2 inc AC ± 180° . (21.)
If we compare (20) and (21) with (19) we find that the last two terms of
(17) have the same inclination . Therefore we can apply the third canon (t o
(17)), that is:
gr(AB • AD)
inc(AB - AD)
= (grCB) 2 + (grAC) 2
= (2 inc AC ± 180°) ± 180° = 2 inc AC.
(22)
202
P. Freguglia
MfM 46: 2
But since inc AC ± 90° = inc DB we have that 2 inc AC + 180° = 2 inc DB
or 2 inc AC = 2 inc DB + 180° ; from which :
and
inc(AB AD) = inc AB + inc AD = 2 inc DB
inc AB + inc DA + 180° = 2 inc DB ± 180°
± 180 °
so that :
inc AB + inc DA = 2 inc DB .
(23)
We also have :
AD + DB -- AB
or
BA + DB + AD r- O .
(24)
Given (23) and (24), we can apply the fourth canon, from which we conclud e
that :
gr AB = gr AD .
(25 )
If we substitute (25) into (22) we have :
( gr
AB ) 2 = ( gr CB ) 2 + ( gr AC) 2
which is Pythagoras's theorem .
From the technical viewpoint this proof is different from the Euclidean proof ,
but it is based on the same principles (see definition of equipollence and the
proof of the fourth canon) .
References
Argand, J .R . 1806, Essai sur une manière de représenter les quantités imaginaires dan s
les constructions géométriques, Paris, M .me veuve Blanc .
Bellavitis, G . 1832, "Sulle quantità immaginarie . Risposta alle osservazioni del Prof.
Crones", Verona , Poligrafo.
Bellavitis, G . 1832b, "Sopra alcuni teoremi di Geometria", Annali Regno Lombard o
Veneto, vol . II .
Bellavitis, G . 1843, "Soluzioni grafiche di di alcuni problemi geometrici trovate col metodo
delle equipollenze", Memorie Istituto Veneto di Scienze, Lettere ed Arti, vol . I .
Bellavitis, G . 1847, "Saggio sull'algebra degli Immaginari", Venezia, Atli Istituto Venet o
di Scienze, Lettere e Arti, tomo VI .
MfM 46 :2
20 3
Bellavitis, G . 1854, "Sposizione del metodo delle equipollenze", Modena, Memorie Società
Italiana, vol . XXV .
Bellavitis, G . 1854b, "Sopra un algoritmo del Grassmann per esprimere gli allineamenti ,
e sull'ordine e la classe del luogo geometrico dei punti o dell'inviluppo delle rette
soggette ad una legge di allineamento", Atti Istituto Veneto di Scienze, Lettere e
Arti, tomo VI, s .Il .
Bellavitis, G . 1858, "Calcolo dei Quaternioni dell'Hamilton e sue relazioni con il metod o
delle Equipollenze", Venezia, Atti Istituto Veneto di Scienze,Lettere e Arti, tomo
III .
Bellavitis, G . 1876, "Sull'origine del metodo delle equipollenze", Venezia, Memorie Istitut o
Veneto di Scienze, Lettere e Arti, vol. XIX .
Buée, A . 1806, "Mémoire sur les quantités imaginaires", London, Trans . Roy . Soc ., 96 .
Canepa, G . 1992, "Le carte Bellavitis", Appendice B in P .Freguglia Dalle equipollenze ai
sistemi lineari. Il contributo italiano al calcolo geometrico, Urbino, Quattroventi .
Canepa, G . 1994, "Le carte di Bellavitis", VV .AA . . Le scienze matem,atiche nel Venet o
dell'Ottocento, Venezia, Istituto Veneto di Scienze, Lettere ed Arti .
Canepa, G ., Freguglia, P . 1991, " Alcuni aspetti della corrispondenza Giusto Bellavitis Angelo Genocchi", VV . AA. Angelo Genocchi e i suoi interlocutori scientifici ( a
cura di A .Conte e L .Giacardi), Torino, Deputazione subalpina di storia patria .
Carnot, L . 1803, Géométrie de Position, Paris, J .B .M .Duprat .
Culmann, K . 1880, Traité de Statique Graphique, (trad . par G .Glasser, J .Jacquier ,
A .Valat), vol .I, Paris, Dunod .
Français, J .F . 1813, "Nouveaux principes de Géométrie de position, et interprétation
géométrique des symboles imaginaires", Ann . de Math . de Gergonne, t . IV .
Freguglia, P . 1991, "I fondamenti dell'algebra degli immaginari secondo Giusto Bellavitis" ,
Atti delle `Giornate di storia della matematica', Cetraro 8-12 . IX . 1988, Rende ,
EDITEL .
Freguglia, P. 1992, Dalle equipollenze ai sistemi lineari . Il contributo italiano al calcol o
geometrico, Urbino, Quattroventi .
Freguglia, P . 1994, "Il calcolo delle equipollenze di Giusto Bellavitis", VV .AA. Le sc . mat .
nel Ven . dell'Ott ., op . cit .
Hoüel, M .J . 1869, Sur le calcul des équipollences, méthode d'analyse géométrique de
Messieur Bellavitis, Paris, Gauthier-Villars
Hoüel, M .J . 1874, Théorie élementaire des quantités complexes, Paris, Gauthier-Villars .
Laisant, C .A . 1874, Exposition de la méthode des équipollences par Giusto Bellavitis ,
traduit de l'italien, Paris, Gauthier-Villar s
Laisant, C .A . 1887, Théorie et applications des équipollences, Paris, Gauthier-Villars .
MfM 46:2
20 5
Hypercomplex Numbers in the Work of Caspa r
Wessel and Hermann Günther Grassmann :
Are there any Similarities ?
Karl-Heinz Schlote *
1 Introductio n
At first glance one might argue that it is somewhat peculiar to compare th e
work of Caspar Wessel and Hermann Günther Grassmann with regard t o
hypercomplex number systems . In fact, it is quite apparent that Grassman n
never heard of Wessel's paper "Om Directionens analytiske Betegning" (O n
the analytical representation of direction) 1 , and reading that title as well as
the titles of most of Grassmann's papers we would not expect a contribution to hypercomplex number systems . However, a second glance at thei r
lives and work already reveals that there are some formal similarities . Both
Wessel and Grassmann had rather isolated positions in the scientific community and their ideas and findings were only appreciated after considerabl e
delay. Until now there are only a few points of resemblance in the living cir cumstances of these two mathematicians . Therefore, this article examine s
whether there are similarities in the contributions of Wessel and Grassman n
to the theory of hypercomplex numbers .
* Sächsische Akademie der Wissenschaften, Postfach 100440, D-04004 Leipzig, Ger many
' In the following the English translation [Wessel, 1999] of Wessel's essay is used .
206
K .-H. Schlote
MfM 46 : 2
2 The contribution of C . Wessel to the theory of
hypercomplex numbers
In his publication "On the analytical representation of direction" C . Wesse l
aimed at developing a geometrical calculus for calculation with directe d
magnitudes, i .e . in modern terms - for calculations with vectors . In the
introduction he stated : "The present attempt deals with the question o f
how to represent the direction analytically, or, how one ought to expres s
straight lines, if from a single equation in one unknown line and some give n
lines one is to be able to find an expression representing both the length
and the direction of the unknown line ." ([Wessel, 1999], 103 )
Thus, Wessel had to give an appropriate definition of operations with directed magnitudes . While working on this, he was led to fundamental con siderations on the character of these operations . He clearly expressed that
the new magnitudes differed substantially from the ordinary numbers an d
that the operations with the former had to be redefined . He solved this problem by generalising from the operations known for calculating with natural
numbers . In other words, he wanted to expand the domain of applicability
or
in modern terms the domain of definition of these operations . At
the same time he tried to maintain as many properties of these operations a s
possible . In this sense his approach is comparable to the ideas developed i n
the English algebraical school, and in particular it reminds us of Peacock' s
principle of permanence of form . However, Wessel's work only marked th e
beginning of this development .
Late in the 18th century, several mathematicians tried to work out an exact
foundation for the number system, above all, to incorporate negative an d
imaginary numbers into the system . During their studies they recognised
basic properties of calculations with numbers, i .e . those properties from
which all other rules for numerical calculations could be derived . However,
basic rules for number systems and calculation with numbers had yet to b e
abstracted and, in consequence, the principle of permanence for operation s
with abstract magnitudes likewise remained unknown . We can trace thi s
process in the work of Wessel . His general notion was to speak of propertie s
for calculations with numbers, and he stressed for the operation of addition
that the added terms can be permuted and that multiplication and additio n
are distributively connected . Of course he did not use the terms commuta-
MfM 46 :2
Wessel and Grassmann on Hypercomplex Numbers
20 7
tive and distributive, since they were only coined several years later . 2 At
nearly the same time, C .F. Gauß, working on binary quadratic forms (cf .
[Gauss, 1801]), pointed to the fact that commutativity and associativit y
cannot be taken for granted for connections in domains of general mag nitudes . These properties must be proven as specific in every connection .
This clearly demonstrates the basic problem of that process : To be able
to investigate a system of magnitudes which includes the realm of positiv e
real numbers or contains magnitudes which are different from numbers, on e
has always to decide which rules of connections are the basic ones . In othe r
words, one has to depict which rules of combination determine the algebrai c
structure of the realm of magnitudes . Therefore, Wessel's statement to the
effect that the rules of connection traditionally used in numerical realms ar e
maintained, was definitely not a trivial one . At that early stage of develop ment it represented a first significant insight into the structure of numerica l
domains .
Shortly after Wessel had published his essay, the now well known propertie s
associativity, commutativity and distributivity were abstracted and becam e
standard conditions in considerations about numerical realms . However, it
should be underlined that this process of abstraction was not only due t o
investigations on number systems (cf. [Novy, 1973], Chap . 4 & 5) .
A second important feature of Wessel's work should be considered . Wessel
chose two directions, first the real axis and a second axis perpendicular t o
the first . He introduced the units 1 and e, corresponding to those directions ,
and made clear that all other magnitudes, i . e. every line segment in the
plane, can be represented by a linear combination of these units . Wessel
derived the multiplication of the units with each other including -1 and -e
as units . At this stage he remarked that e = V-1, and therefore comple x
numbers are contained in his investigation as a special case . The geometrical
representation of the units, together with the rule that the amplitude of th e
product of two line segments is equal to the sum of the amplitudes of it s
factors, formed the basis for Wessel's determination of the multiplication o f
any two line segments . He stated that the product was taken componentwis e
and was determined unequivocally by the products of the units .
That concept of unit was geometrically motivated but one should not minimize its stimulating impact on algebra . The same is valid for Wesel`s insigh t
2 The terms "distributive" and "commutative with each other", for instance, wer e
coined by F .-J . Servois in a paper about the theory of operators in 1814 . [Servois, 1814]
208
K.-H. Schlote
MfM 46: 2
that the product was determined by the products of the units . The geometrical motivation caused Wessel to extend his method to space, hence t o
extend multiplication to three-dimensional magnitudes . Trying to realiz e
this, Wessel introduced a further unit rl, whose square is equal to -1, which
was the same condition as for E . Geometrically, the new unit represented a
direction which is perpendicular to the plane generated by the units 1 an d
E . Then Wessel defined a specific connection, though not a multiplicatio n
as Wessel remarked, since it was applicable only to some but not to al l
of his three-dimensional magnitudes . The connection pointed to a strong
geometrical influence and was constituted in such a way that products o f
the units e and rl were avoided . Thus, Wessel had targeted a problem, the
solution of which resulted in the emergence of the first system of hyper complex numbers . However, the solution of the problem was not given b y
W.R. Hamilton until October 1843, after more than 10 years of intensive
investigations . This was the birth of the quaternions and marked a clima x
in the history of hypercomplex number systems . His own restricted calculus
was obviously sufficient for Wessel to reach his objectives .
Apart from the idea of extending and generalizing the complex numbers t o
space, most of Wessel's results mentioned above were also stated by othe r
mathematicians who directed their efforts toward the foundation of imaginary numbers .
Concluding the discussion on Wessel's work, two aspects should be emphasized . First, Wessel's essay was the first publication giving an exac t
treatment of the geometrical representation of complex numbers and, o n
this basis, of calculation with them . Second, Wessel situated his investigations in the general framework of searching for a geometrical calculus . Thi s
forced him to think about the foundation of the connections of the ne w
magnitudes and the properties of the connections . Furthermore, his search
for a geometrical calculus was an important impulse to extend his derive d
method to space . Although Wessel was not entirely successful in this at tempt, the description of which new problems emerge in the calculation s
with three-dimensional magnitudes was of the utmost importance for further developments .
In this context it should only be noted that Wessel prepared and develope d
basic elements of a vector calculus . It remains speculative whether Wesse l
had been aware that he followed the tradition of Leibniz . The commenta-
MfM 46 :2
Wessel and Grassmann on Hjpercomplex Numbers
20 9
tors on the French translation of Wessel's article, H . Valentiner and T .-N .
Thiele, did not refer to this tradition . They only indicated that the geometrical representation of imaginary numbers as well as calculations with the m
are of relevance and established a relation to the quaternions of Hamilton .
3 The contributions of H . G . Grassmann to th e
theory of hypercomplex number s
Many of Grassmann's ideas, developed about 50 years later, can be seen as a
continuation of Wessel's results . This supports the assessment that Wessel' s
essay had already uncovered important problems for future research .
At first some biographical data on Grassmann should be given as far as
they concern his research discussed below . Hermann Günther Grassmann
was born in Stettin on April 15th, 1809. Starting in 1827 he studied theology for 6 semesters at the University of Berlin . During this time he als o
attended lectures in philosophy and philology . Thus, Schleiermacher's idea s
on dialectics, with which he became acquainted during his studies, as wel l
as the philosophy of Kant, had a strong impact on Grassmann's work . After
he had returned to Stettin in 1830, he devoted his efforts to the intensiv e
private study of mathematics and physics as well as theology . Finally, he
qualified to teach these subjects with a career as a teacher in view . Between 1831 and 1840 he passed teaching examinations for the gymnasiu m
in Stettin, where he worked as a teacher from 1835 onwards . All his effort s
to obtain a professorship at a university failed even though he publishe d
his theory of extensions ("Die lineale Ausdehnungslehre . . . ") in 1844 and
contributed to the advance of mathematics in several books and articles . .
Only a few years before Grassmann's death did some mathematicians com e
to appreciate his ideas, at which point he received some acknowledgement i n
the scientific community . An exception was his philological studies on San skrit which were immediately appreciated by scientists . Grassmann died on
September 26, 1877 .
The mathematical ideas of Grassmann developed slowly . It took severa l
years before he had brought these into a programme which he attempte d
to realize in cooperation with his brother Robert . One source from whic h
Grassmann derived aspects of his research was his occupation as a teacher .
210
K.-H. Schlote
MfM 46:2
In this context he undoubtedly had to think about the structure of mathematics as a science and realized that it was necessary to give an exac t
foundation and construction of mathematics . Thus, at the beginning of the
1830s he was led to the concept of a new addition of directed line segments .
This was nothing less than the addition of vectors . From a retrospective
point of view, the description Grassmann gave was very similar to Wessel' s
explanations in his essay . However, daily duties at school prevented Grass mann from working out his new method in more detail . Moreover, he ha d
not yet been convinced of the efficiency of the new method . His further research was stimulated by publications of his father, who had investigate d
products in geometry in particular in the book "Raumlehre" [Grassmann,
1824] . Grassmann developed his father's approach further, defined a produc t
of directed line segments which corresponds to the modern vector product ,
and combined the product with the addition of vectors established som e
years ago . This last step seems to be very natural, but it was a decisiv e
one . Now he suspected that he had created the beginnings of a new efficien t
method . It was already during this early stage that Grassmann discovere d
that the new multiplicative operation was non-commutative . In fact thi s
"curious result" as Grassmann named it initially embarrassed him, nevertheless it was not a sufficient reason for him to reject this multiplicatio n
([Grassmann, 1894] Band I, Theil 1, 8) . Once again, he did not have the
time to elaborate his findings . He returned to his method only in 1839 whe n
he worked on the theory of ebb and flow of P.S . Laplace during his teaching
examinations . He simplified and improved the representation of Laplace' s
theory and concluded :
Thus I feel entitled to hope that in this new analysis I have found th e
only natural way in which mathematics should be applied to nature ,
and likewise that in which geometry should be treated . . .
([Grassmann, 1844a], 10) 3
Within four years he elaborated the calculus completely and published th e
book : "Die lineale Ausdehnungslehre . . . " Nobody denies that Grassmann
3 "Durch diesen Erfolg nun hielt ich mich zu der Hoffnung berechtigt, in dieser neuen
Analyse die einzig naturgemässe Methode gefunden zu haben, nach welcher jede Anwendung der Mathematik auf die Natur fortschreiten müsse, und nach welcher gleichfalls die
Geometrie zu behandeln sei, . . . " ([Grassmann, 1894] Band I, Theil 1, 8) . The quotation
follows the English edition of Grassmann's theory of extension . (I am grateful to Dr . G.
Schubring who sent me a copy of a part of that edition .)
MfM 46:2
21 1
presented important new results on geometry but here we shall conside r
only his ideas on hypercomplex numbers . One would not expect a relatio n
between geometry and hypercomplex numbers in a book entitled "Linear e
Ausdehnungslehre" (Linear extension theory) . The results on hypercomplex
numbers are basically due to the fact that Grassmann treated problems i n
a very abstract manner . He started the book with a general philosophica l
introduction and defined mathematics as a formal science that examined
objects established by pure thought . He intended to give a uniform forma l
and constructive development of mathematics . 4 For that, he chose the concept of form or of "thought-form" as a starting point and began with genera l
considerations on connections of abstract magnitudes . In the end he justified his discussions for fitting in the new science too . Grassmann regarde d
the properties of connections as those rules which are valid for all form s
and with it for all parts of mathematics . This resulted in an emphasis o n
the rules of connections and a removal of the connected magnitudes fro m
the world of concrete particular objects . In abstracting from the concret e
realization of the connections in several parts of mathematics, Grassman n
performed an important step and tried to grasp the structure determine d
by the connections . This demonstrates Grassmann's insight that it was necessary to abstract from the peculiarities of the connected objects to arriv e
at common structural properties . The significance which Grassmann attached to the rules of connection was documented, among other things, b y
his division of mathematics into four branches and his definition of mathematics as a "science of the connection of magnitudes", both based on th e
concept of connections . This opinion was still expressed in more detail i n
the book "Formenlehre" (Theory of forms) [Grassmann, 1872] written b y
Robert Grassmann in cooperation with H . G . Grassmann . There Robert
Grassmann wrote :
The theory of forms or mathematics consists of five branches, a genera l
one, the science of magnitudes, and four particular branches .
1) The science of magnitudes, both the first and general branch o f
the theory of forms, tells us about the connection of magnitude s
which are a common part of all branches of the theory of form s
([Grassmann, 1872], 11) 5
4 The philosophical background is discussed in [Lewis, 1977] .
5 The following quotations were translated by the author . The original German version
is given in the notes . "Die Formenlehre oder die Mathematik zerfällt in fünf Zweige, einen
212
K .-H.
Schlote
MfM 46 : 2
The "theory of forms" was part of the programme for a new foundation o f
mathematics pursued by Robert and Hermann G. Grassmann . The above
quotations also reflect H .G. Grassmann's opinion towards this aspect . Moreover, these quotations express an extended concept of magnitudes elaborated by Grassmann in his research . In 1844, in the "Ausdehnungslehre" ,
Grassmann refused the characterization of mathematics as a science of magnitudes because the concept of magnitudes was too narrow and as a consequence restricted to numbers . Step by step Grassmann presumably recognized that his definition of a magnitude as "everything which was receive d
via thinking" ([Grassmann, 1894] Band I, Theil 1, 24) can be interpreted
in a much more abstract manner than he had done in the book "Ausdehnungslehre" .
After he had divided mathematics into four branches and determined th e
position of the theory of extensions as a new branch of mathematics, Grass mann analysed the properties common to all connections in the genera l
theory of forms . At first he introduced the concept of equality, then h e
treated the connections . He supposed the following as a principle : Any tw o
magnitudes are connected and the result of their connection has to be unambiguous . If more than two magnitudes should be connected, this will b e
carried out by successive operations on two magnitudes .
In this context Grassmann addressed the important problem of whethe r
a set of magnitudes is closed under the connection . This property is ver y
conclusive for defining a mathematical structure . Grassmann did not discuss
the question in an abstract manner . That is understandable, since it was
not his purpose to define and analyse mathematical structure . Nevertheless
it can be seen that Grassmann discerned this problem as an important
one . In most of the cases he investigated, Grassmann considered the syste m
of magnitudes as closed . He derived this conviction from the manner i n
which magnitudes were generated . But in some cases he could not dra w
this conclusion : for instance, when he connected extensive magnitudes by
multiplication he got extensions of a higher degree . The result was again a n
extensive magnitude of course, but the multiplication could formally yiel d
a magnitude (or extension) of any degree . In such cases Grassmann di d
allgemeinen Zweig, die Grössenlehre und vier besondere Zweige . 1) Die Grössenlehre, der
erste oder der allgemeine Zweig der Formenlehre, lehrt uns die Knüpfungen der Grösse n
kennen, welchen allen Zweigen der Formenlehre gemeinsam sind, . . . "
MfM 46 :2
21 3
not answer the question of closure . Only when the system is reduced to a
finite number of units of first degree the extensions of higher degrees becom e
closed in a natural way . (Every product with more than n factors is equa l
to zero .) As examples we mention, first, the combination of progressive an d
regressive product to the "product with regard to a principal domain" an d
second, the investigation of sub domains in a larger domain of magnitudes (o r
what we would call, in modern terms, the properties of a vector subspace) .
In the first case the product was defined in such a way that the result wa s
contained in a domain determined in advance . To describe the second cas e
Grassmann stated :
Since every system which contains some magnitudes has to contain
also all magnitudes depending on the former ones and also the oute r
product of them, i . e. the whole system which is determined by the m
([Grassmann, 1894], Band I, Theil 1, 206f) 6
Again, both instances clearly showed the significance which Grassmann attached to a closure of the system .
In a system of magnitudes with one connection, the permutation of th e
members and a change of the brackets are the only two possible operation s
which could be carried out without changing the forms or the result of th e
connection, respectively. This would mean that the connection satisfies th e
commutative and the associative laws . A connection which satisfies bot h
laws is called a simple connection . After that Grassmann treated the so called resolution of a given connection or the inverse operation, which h e
called an analytical connection . The definition of this resolution is so general that a non-commutative connection can be considered initially, whic h
Grassmann also termed a synthetic one . However, Grassmann restricted his
investigations immediately to simple synthetic connections and asked fo r
conditions under which both operations can be permutated and brackets ca n
be added or omitted . He recognized that the analytical connection can b e
ambiguous in some particular cases . In cases where the synthetic connectio n
was simple and the accompanying analytical connection was single-valued ,
he called the former addition and the latter subtraction . Later Grassman n
additionally introduced an indifferent and a purely analytical form . In modern notation these correspond to an identity element and an inverse element ,
6
"Da nun jedes System, welches gewisse Grössen enthält, auch sämmtliche von ihne n
abhängige Grössen, das heißt das ganze durch sie bestimmte System, also auch das äusser e
Produkt jener Grössen, enthalten muss, . . . "
214
K.-H. Schlote
MfM 46: 2
respectively. The indifferent form was obtained by an analytical connectio n
of two equal forms, the purely analytical one by an analytical connectio n
of an indifferent form with the initial one . Notably, Grassmann explaine d
that if the analytical connection is assumed to be single-valued, then th e
definition of the indifferent form is independent from a special choice of th e
form . This can be seen as a first sign of a new algebraic way of thinkin g
and as a step towards defining objects by means of equivalence classes . In
addition to this prospective aspect one can find a remarkable weakness i n
this definition . An identity element had only been defined, if the inverse operator existed and was unambiguous . Thus, a ring with zero-divisors woul d
never have an identity.
When a second synthetic connection existed, Grassmann regarded the mutual relation as the main problem . He gave an argumentation from which a
distributive combination of both operations derived . If the operations wer e
distributively connected and the first one was an addition, then he called
the second one a multiplication . A set of magnitudes supplied with two
such connections constitutes a ring, but Grassmann neglected this structural point of view. There is no one position in Grassmann's work wher e
he alluded to the definition of a new mathematical object, a new algebraic
structure . Although he had mentioned all properties which are neccessary
for an axiomatic characterization of a commutative group or a ring, he neve r
realized this step . The possibility of defining a set of magnitudes having one
or two connections as a new mathematical entity was overlooked by Grass mann. He identified the concept of connection in a system of magnitude s
as the most decisive one, and the different determinations of the connection as presenting possibilities for the analysis of the various systems o f
magnitudes, but he did not recognize that the systems represent differen t
mathematical objects with regard to the different algebraic structures . This
gives a clear idea of the extent to which Grassmann contributed to the genesis of modern algebra and his limitations . He established the opportunity
for defining abstract structures, but he was not in a position to conside r
abstract algebraic structures . However, this does not diminish Grassmann' s
achievement . How far he was ahead of his contemporaries can be seen fro m
the fact that according to him, a system of magnitudes carrying an additio n
has to fulfil certain conditions which implicitly suppose the existence of a
certain structure .
Grassmann concluded his general theory of forms with some remarks on di-
MfM 46:2
215
vision as the analytical connection to multiplication, but he did not presen t
any new ideas . Again he stated the necessity of defining two analytica l
connections as a consequence of the non-commutative multiplication . After
these general considerations on connections, Grassmann developed his elaboration of the theory of extensions in which he applied his ideas in variou s
ways . His description was verbose and difficult to understand . Grassmann
was not satisfied with the level achieved by his theory, and he therefor e
explained some of its details in further articles . Finally, he published a re vised edition of the "Ausdehnungslehre" in 1862 "in Euclidean form" (that
is, in stronger Form), which pointed out the formal aspects and completel y
omitted the philosophical considerations . This shows that Grassmann di d
not extend the domain of research chosen earlier, but within it he expresse d
his ideas more clearly and gained a better mastery of the methods .
Within a relatively short time he had reached a new grasp of the structur e
of his theory and an improved mastery of it . This can be derived from a
comparison of the book "Ausdehnungslehre" (1862) with his textbook o n
arithmetic from 1861 and the article "Sur les différents genres de multiplication" from 1855 . What basic changes can be traced ?
Grassmann started with the concept of unit and a system of units as a
basis of a system of magnitudes or extensive magnitudes, respectively. An
extensive magnitude was defined by
al e' + a2e2 + . . . _ E aiei
with ai real numbers, called numbers of derivation (Ableitungszahlen) by
Grassmann and {ei} denoting a system of units . The sum above was of
course a finite one, but Grassmann did not mention it in any case, and usually it is also possible to think of an infinite one . He defined the addition o f
extensive magnitudes componentwise and had then to define the multipli cation of an extensive magnitude with a scalar, usually a real number . H e
determined it in a formal way b y
~ * E aiei, _ (E a i e i) * ß = E(ßai) e i •
Although he regarded only real numbers as the domain of coefficients, i t
is possible to choose any abstract coefficient field without difficulties . The
multiplication satisfies the following rules (cf . [Grassmann, 1894], Band I,
K.-H. Schlote
21 6
MfM 46:2
Theil 2, 14)
a*ß
(a+b) *
a*1
a* 13
(a*ß)*-y = a *(ß'y) ,
* a,
a*ß+b*ß,
a* (ß+y) = a*ß+a*y ,
a,
0 if and only if either a = 0 or ß = O .
In an attempt to compare Grassmann's and Wessel's work, a short overvie w
of some geometrical aspects of Grassmann's theory of extension should b e
given at this stage . At first Grassmann had selected all properties which h e
would need for an abstract definition of a linear vector space as a new mathematical object and again he omitted this level of abstraction . He performe d
his calculations in a domain or system of extensive magnitudes without con sidering it as a mathematical object in its own right . Nevertheless, he derive d
important results for vector spaces . Thus he described the linear dependenc e
of extensive magnitudes as standing in a numerical relation to each other .
In the article mentioned above he also supposed the linear independence o f
the relative units which form the basis of the extensive magnitudes. All in
all Grassmann included the following theorems of linear vector spaces in hi s
book :
A domain (or system) of nth order is generated by exactly n independent elements . The concept of a basis is not used by Grassmann . A
domain of nth order corresponds to an n-dimensional space.
The theorem of exchange, later also called the fundamental replace ment theorem of Steinitz, considering the possibility of replaceing a
certain set of q vectors from among the vectors a l , a2 , . . . , ap which
span a vector space L by q linear independent vectors b i , b2 , . . . , bq
of L (q < p) .
The theorem on the dimension of the sum of two linear subspaces
dim(E + F) + dim(E n F) = dim E + dim F.
Even in this context Grassmann appears as an intellectual successor of Wes sel, and his initial findings on vector calculus . It is common knowledge that
some of the above stated theorems of linear spaces also play an importan t
role in algebra, which was probably first announced by R . Dedekind in his
famous 10th supplement to Dirichlet's "Lectures on number theory" . However, Grassmann could not have foreseen those aspects .
MfM 46:2
Wessel and Grassmann on Hypercornplex Number's
21 7
There is still another problem on a less abstract level which has to be analysed with regard to Grassmann's contribution to the theory of hypercomple x
numbers . It concerns the product of two extensive magnitudes . The most
detailed description is given by Grassmann in the already mentioned articl e
"Sur les différents genres de multiplication" written in a priority claim mad e
against A.L. Cauchy.
In accordance with his general remarks in the "Ausdehnungslehre" of 1844,
Grassmann defined the multiplicative connection of two extensive magnitudes by
i arer * >
ßs
e
s = > a
r38 (er *
e
s)
The distributive law was tacitly assumed and no further restrictions wer e
made . This was completely correct since he wanted to define a multiplicative
connection of magnitudes which were already connected by addition . In
this case both connections have to be combined by the distributive law .
Furthermore he stressed that the product of the two relative units has t o
be regarded as a new unit of second degree, or of nth degree if n unit s
are multiplied . This point of view was very general, even too general, t o
derive some results or to apply his considerations on a geometrical problem ,
for instance . Thus Grassmann had to make further specifications to th e
product (e r * e s ), and therefore he introduced the so-called "equations of
determination" (Best immungsgleichungen) . He stated :
Since the product of two extensive magnitudes is according to the definition again either an extensive magnitude or a numerical magnitud e
(Zahlgröße), it must be numerically derivable from a system of units .
This definition does not explain which system of units it should be an d
how the products [e r * e5 ], of which these products are composed, ar e
numerically derivable from these units . If a particular product shoul d
be determined in detail, the necessary determinations on this syste m
of units and these derivations have to be made . ([Grassmann, 1894] ,
Band I, Theil 2, 28) 7
'' "Da das Produkt extensiver Grössen nach der Erklärung wieder entweder eine extensive Grösse oder eine Zahlgrösse ist so muss dasselbe ( . . . ) aus einem System von Einheiten numerisch ableitbar sein . Welches dies System von Einheiten sei, und wie aus ihne n
die Produkte [e, .*e 5], aus denen jenes Produkt zusammengesetzt ist, numerisch abzuleiten
seien, darüber sagt die Definition nichts aus . Soll also der Begriff eines besonderen Produktes genau festgestellt werden, so müssen noch über jenes System von Einheiten un d
über diese Ableitungen die nötigen Bestimmungen getroffen werden ."
218
K.-H. Schlote
MfM 46 :2
Each system of equations of determination defined a specific kind of product . Grassmann realized that different products result in different system s
of magnitudes and therefore the kinds of products or the equations of determination furnish a decisive distinguishing feature for systems of magnitude s
and a possibility to classify them . To master this problem would require a
complete classification of hypercomplex number systems, and thus it is no t
surprising that Grassmann solved these problems only in particular cases .
His choice of equations for a further determination of the products wa s
clearly influenced by geometrical ideas, since, as he said: "nature calculates
in a geometrical manner" ([Grassmann, 1894], Band III, Theil 1, 33) . But
many factors which cannot be analysed in this article have to be borne i n
mind. 8 Only two facts should be stated . Firstly, Grassmann did not take
into account other ideas of his time such as Hamilton's quaternions, maybe
as a consequence of Grassmann's isolated position at his grammar schoo l
at Stettin . Secondly, he restricted the equations of determination to thos e
in which all products have the same number of factors, mostly on products comprising only two factors . Grassmann gave no explanation of thi s
very restrictive condition and his reasons remain a mystery for us . Mayb e
his statement that only connections of two objects will be considered ha d
an impact on the decision . As a consequence, the reduction of a produc t
to a linear combination of units was excluded since in this case a produc t
of two units would be equal to a sum of basic units (with some numerical
coefficients) . The same was applied to relations like the associative law fo r
instance . Only as far as the associative law was concerned did Grassman n
break this rule and he considered some associative systems, too .
In order to select from the infinity of possible special multiplications thos e
which seemed to be useful in science, Grassmann formulated three genera l
properties. He formulated these properties as an invariance of the equations
of determination towards some transformations of the units of the basis .
These transformations were as follows :
(1) The change of the sign of a relative unit or the change of the signs o f
two units and the permutation of those units with each other .
(2) The substitution of any two units by two extensive magnitudes, whic h
are formed by those units but which are different from those units as
8 For an extended discussion of that problem as well as Grassmann's contributions t o
the theory of hypercomplex number systems cf . [Schlote, 1985] .
MflVI 46 :2
21 9
well as a multiple of them.
(3) The substitution of any unit by any extensive magnitude formed b y
these basic units .
The multiplications satisfying the first, the first as well as the second, an d
all properties were called symmetrical, circular and linear, respectively. Despite the restrictions, Grassmann distinguished between sixteen differen t
types of symmetrical multiplications and analysed them . Applying thes e
examples he was able to demonstrate how he perceived the realization o f
his principles . Among others, he deduced the equations of determination fo r
all these different kinds of multiplication . These equations were obtained b y
a combination from the following ones :
er*es=es*er ,
er*es+e s *er =0,
el*el =e2*e2= . . =en*en ,
el * el + e2*e2+ . . .+en*en=0 .
Grassmann continued by separating the circular multiplications and, finally,
the linear multiplications from the deduced 16 kinds . The four linear one s
he characterized as being of the utmost importance for analysis, geometry,
mechanics and physics . Among the linear products there are some of particular interest: the algebraic product which generates a commutative algebr a
(in general, the algebra is assumed to be associative) ; and the combinatoria l
product or outer multiplication from which a non-commutative algebra wa s
derived (later called a Grassmann algebra by W.C. Clifford) . The equation s
are
er * es = -es * e r
and
er = O .
The circular multiplication also encloses some further remarkable product s
which were not linear, the inner product or inner multiplication and the com plex one . The first was known from its geometrical applications and Grass mann refered to his "Ausdehnungslehre" and his "Geometrische Analyse "
where he extensively used this product . The latter one was a generalizatio n
of the multiplication of complex numbers, a fact which was also pointed ou t
by Grassmann . The defining equations wer e
er * e s = 0 and e l * e l = e 2
er* es =e s *er
* e2 =
= en * en , r ~ s, respectively
and el*el+e 2 *e2 + . . . + en *en=0 .
220
K.-H. Schlote
MfM 46 : 2
Again, there is an analogy to Wessel's work . Both Wessel and Grassman n
referred to a useful application of their ideas in the theory of functions .
However, Wessel's remark was only a very vague suggestion . It is difficul t
to estimate how far they went in this direction or whether they had concrete important results in mind . In the final analysis, a lasting value ca n
be seen in the attempt to stimulate some parts of mathematics which were
to constitute a dominating feature in the development of mathematics i n
the nineteenth century : the theory of complex functions and the theory o f
functions of several variables, respectively.
4 A comparison of Wessel's and Grassmann' s
work
The analysis of Wessel's and Grassmann's work on hypercomplex number s
has revealed several similarities . With regard to their mathematical researc h
they had a comparable fate . Their ideas were not appreciated by their contemporaries, and the importance as well as the implications of the ne w
methods was underestimated . Whereas Wessel published his article as a
man of mature age, Grassmann was a young man at the beginning of hi s
career . The failure of Wessel's essay to find an audience did not influence hi s
professional career, while Grassmann suffered from the lack of acceptanc e
of his ideas . His professional career was also affected by this . During his life
he fought for the diffusion of his method and for an appropriate assessmen t
of it but only with little success near the end of his life .
From the mathematical point of view the investigations of both mathematicians can be integrated into the Leibnizian programme to look for a geomet rical calculus . At first Wessel and Grassmann did not see themselves in th e
Leibnizian tradition . Later however, Grassmann perceived it and considere d
the theory of extensions as a solution of the Leibnizian task . He drew up the
theory from this point of view in the article "Geometrische Analyse" (Th e
geometrical analysis) [Grassmann, 1847] . Of course, there were many other
factors which had an effect on the work of both mathematicians and whic h
were quite different . Nevertheless, the search for a geometrical calculus wa s
a common characteristic . In addition, there are some similarities in the applications of their methods they searched for, in geometry for instance, but
also in other parts of mathematics . In many ways Grassmann's work can be
MfM 46 :2
22 1
seen as a continuation and broadening of Wessel's ideas without any direc t
reference of Grassmann to Wessel's essays . The topics they discussed were o f
interest at their time, and both men provided a number of solutions whic h
were ahead of their time . Their mathematical achievements had erroneousl y
been marginalized for a long time, but they were never outsiders in mathematical research . On the contrary, Caspar Wessel and Hermann Günther
Grassmann had a profound knowledge and understanding of mathematica l
research problems . Their contributions to mathematics were durable, and
they are still appreciated in our time .
References
Gauss, Carl Friedrich, 1801 : Disquisitiones arithmeticae . Leipzig 1801 (Untersuchunge n
über höhere Arithmetik. Deutsche Ubers . H . Maser . Berlin 1889 )
Grassmann, Hermann Günther, 1844 : Die lineale Ausdehnungslehre ein neuer Zweig de r
Mathematik . Leipzig 1844 . In : [Grassmann, 1894], Band I, Theil 1 .
Grassmann, Hermann, 1844a : A New Branch of Mathematics . The Ausdehnungslehre of
1844 and Other Works. Translated by Lloyd C . Kanneberg . Chicago, La Salle, Ill .
199 6
Grassmann, Hermann Günther, 1847 : Die geometrische Analyse geknüpft an die von Leibniz erfundene geometrische Charakteristik. Leipzig 1847 . In : [Grassmann, 1894] ,
Band I, Theil 1, p .321-399 .
Grassmann, Hermann, 1855 : "Sur les différents genres de multiplication" . Grelles Journa l
für die reine und angewandte Mathematik 49 (1855), p .123-141 . In [Grassmann ,
1894], Band II, Theil 1, p .199-21 7
Grassmann, Hermann, 1861 : Lehrbuch der Arithmetik für höhere Lehranstalten. Berli n
1861 .
Grassmann, Hermann Günther, 1862 : Die Ausdehnungslehre . Vollständig und in strenger
Form bearbeitet. Berlin 1862 . In : [Grassrnann, 1894], Band I, Theil 2 .
Grassmann, Hermann Günther, 1894: Hermann Grafmanns Gesammelte Mathematisch e
und Physikalische Werke, herausgegeben von Friedrich Engel . Leipzig, 1894-1911 .
Grassmann, Justus Günther, 1824 : Raumlehre für die unteren Klassen der Gymnasie n
und für Volksschulen. Zweiter Theil : Ebene räumliche Größenlehre . Berlin 182 4
Grassmann, Robert, 1872 : Die Formenlehre oder Mathematik . Stettin 1872 .
Lewis, Albert C ., 1977 : "H . Grassmann's 1844 Ausdehnungslehre and Schleiermacher s
Dialektik" . Annals of Science 34 (1977), p .103-162 .
Novy, Lubos, 1973 : Origins of Modern Algebra. Leyden 1973 .
Schlote, Karl-Heinz, 1985 : "Hermann Grassmanns Beitrag zur Algebrentheorie" . Janus .
Rev . intern . hist . sci ., méd. pharm . et. techn . LXXII (1985), p.225-255
Servois, François-Joseph, 1814 : "Essai sur un nouveau mode d'exposition des principes
du calcul différentiel" . Ann. Math . Pures et appl. 5 (1814/15)
222
K.-H. Schlote
MfM 46 : 2
Wessel, Caspar ; 1999 : Om Directionens analytiske Betegning . English Translation : On
the Analytical Representation of Direction . An Attempt Applied Chiefly to Solvin g
Plane and Spherical Polygons . Translated by Flemming Damhus . With introductory
chapters by Bodil Branner, Nils Voje Johansen, and Kirsti Andersen . (ed . Bodil
Branner and Jesper Lützen), Matematisk-fysiske Meddelelser 46 :1, Det Kongelige
Danske Videnskabernes Selskab, Copenhagen 1999 . French Traslation : Essai sur la
représentation analytique de la direction . H . Valentiner ; T .-N . Thiele (ed.) . Copenhague 1897 .
MfM 46 :2
22 3
Julius Petersen, Karl Weierstrass, Hermann
Amandus Schwarz and Richard Dedekin d
on Hypercomplex Numbers
Jesper Lützen *
Abstract
Caspar Wessel's paper On the Analytical Representation of Direc was the first Danish-Norwegian contribution to the field of hyper complex numbers . The present paper is devoted to the second Danis h
contribution to this field : the papers by Julius Petersen . These contributions that were marred by flaws and unrecognized repetitions o f
earlier results are analysed in the context of the contemporaneous wor k
of Weierstrass and Dedekind . This brings out the central position of
H .A . Schwarz . As a side issue it is discussed how diverging interpreta tions of a quote by Gauss resulted in a polemic between Weierstras s
and Dedekind .
tion
1 Petersen and Danish Mathematics at the end o f
the 19th Centur y
While astronomy and geodesy flourished in Denmark-Norway around Wessel's time Wessel's own, then overlooked paper, was the only valuable Danish and Norwegian contribution to mathematics . After Erasmus Bartholin
(1625-98) and Georg Mohr (1640-97) Danish mathematics only reached a n
international level around 1880 with the works of H .G. Zeuthen (1839-1920)
on enumerative geometry and history of mathematics as well as those o f
* Department of Mathematics, Copenhagen University, Universitetsparken 5, DK-210 0
Kobenhavn Ø, Denmark
224
J . Lützen
MfM 46 : 2
Julius Petersen (1839-1910), T .N. Thiele (1838-1910) and later J .P. Gram
(1850-1916) and L .V.W .V. Jensen (1859-1925) . Julius Petersen, who was
Zeuthen's former schoolmate and lifelong friend, was a docent at the Poly technic Highschool in Copenhagen from 1871 to 1887 when he got a chai r
at the University of Copenhagen . He was famous (infamous) to Danis h
schoolchildren who almost all used his system of textbooks in mathematics .
Internationally he became known at the time as the author of Methods and
Theories for the Solution of Problems of Geometric Constructions [1866/79] ,
the best selling mathematics book ever written by a Danish mathematician, translated into innumerable languages and republished as late as 1990 .
His book on the theory of equations [1877] and his lectures on mechanic s
[1881,84,87] also enjoyed wide fame . In recent years he has become famous
mainly through his pioneering contributions to graph theory, cryptography
and mathematical economy .
His works were characteristic through their concise, elegant, often geometric intuitive style, which was not overburdened by details and sometime s
passed over major problems without noticing it . The way he did research
had several connected characteristics : 1° he rarely read the works of other
mathematicians, 2° he was a problem solver, not a builder of theories, 3 °
he only pursued an idea a little way (until he had solved his problem) an d
then dropped it . This had the unfortunate consequence that many of hi s
"discoveries" turned out not to be new and his genuinely new ideas ha d
little influence on his contemporaries because he did not pursue them fa r
enough .
In this paper I shall illustrate many of these points in a discussion of Petersen's research on hypercomplex numbers . More information about Pe tersen and his work can be found in [Lützen, Sabidussi and Toft 1992] an d
[Christiansen et al 1992] .
2 Petersen on the Foundation of Mathematic s
Petersen displayed his first interest in hypercomplex numbers in 1883 in con nection with a discussion on the nature of mathematical axioms that too k
place in the Danish journal Tidsskrift for Mathematik, also called Zeuthen's
Journal. Compared to other participants in the discussion who argued tha t
axioms were based on experience or were a priori in Kant ' s sense, Petersen's
view was extremely modern :
MfM 46:2
Hypercomplex Numbers in the 1880es
225
Mathematics chooses its assumptions in an arbitrary way and deduce s
from them what can be deduced in a logical way. It has little scientifi c
importance that the assumptions are chosen for practical reasons wit h
a view to what appears in nature [ . . . ] When I have said that Mathematics can choose its assumptions arbitrarily, it should perhaps b e
added that the assumptions must not contradict each other . [Petersen
1883, 3-4]
Petersen illustrated this view of axioms by introducing a model of a non Euclidean geometry (the so-called Riemannian or single elliptic geometry
in Klein's terminology) . He added the model of hyperbolic geometry published by Poincaré the previous year in Acta Mathematica [Poincaré 1882] ,
but he seems to have been unaware of Beltrami's work as well as of Klein' s
more farreaching work from 1871 on the "so-called non-Euclidean geometry" [Klein 1871] . However, he soon learned about it . Indeed, a highschool
teacher from Schleswig-Holstein, Rudolph von Fischer-Benzon (1839-1911) ,
who over the years translated many books and papers by Petersen into Ger man, also translated this paper and sent it to Felix Klein (1849-1925) fo r
publication in the Mathematische Annalen . Klein did not find much new in
Petersen's paper :
The comparison with Poincaré's geometry is old . The same applies t o
C3 [Petersen's projective model] . The use of the linebundle is also du e
to me as well as the criticism of Legendre's proofs . What is now left fo r
Petersen? Many interesting specific points . Then the exposition of th e
general principles which are indeed creditably clear . [Klein's remark
in the margin of a letter from Fischer-Benzon to Klein, November 6 ,
1886]
In the end Klein decided that the new observations and the clarity of th e
exposition warranted that the paper be printed in the Mathematische Annalen provided that Petersen added references to the relevant literature tha t
he had originally ignored .
3 Petersen's 1885 Paper on the Basic Concepts o f
Algebra
Petersen's 1883 paper also contained a few remarks concerning the basi c
concepts of algebra . He returned to this subject two years later in a talk to
J. Lützen
226
MfM 46:2
the Royal Danish Academy of Sciences and Letters, of which he had becom e
a member in 1879 . The paper was printed in Tidsskrift for Mathematik the
same year [Petersen 1885] . Petersen defined algebra as a theory of sig n
language with the following characteristics :
1) The language must contain signs that designate the elements in on e
or more sets (Petersen used the word "group" in the sense "set" o r
"algebraic structure") . The meaning of the signs is only determine d
when the algebraic theory is applied .
2) The language must contain signs for operations .
3) The language contains an equality sign .
4) The language has certain arbitrarily chosen basic characteristic equations .
This characterization of algebra and Petersen's description of isomorphi c
structures again seem very modern in view of the fact that by 1885 the development of algebra as the science of algebraic structures was hardly visible .
However, it is in fact unclear precisely how generally Petersen considere d
his characterization . As one among the commonly chosen "basic equations "
Petersen mentioned the commutative law : a + b = b + a, but he did no t
introduce the axiomatic definition of a group (in the modern sense of tha t
word) . In his algebra book from 1877 groups were defined as permutation
groups. In fact, in order to illustrate what he had in mind, Petersen di d
not mention any of the axiomatic structures that were taking shape aroun d
this time such as group or field (cf . e.g. [Wussing 1984]), but he turned t o
hypercomplex numbers .
3.1
Complex Numbers
He first introduced ordinary complex numbers in an unorthodox projectiv e
way : Let L be a given straight line and O a given point outside the line .
Petersen now defined the sum C of two points A and B in the half plane
containing O by the construction in figure 1 . He claimed without proof tha t
this definition of addition is commutative and associative and has O a s
a neutral element (called 0) . Moreover, the points of L are infinite in th e
sense that they are not changed if another point is added to them . Althoug h
addition of two points on the same line through O is not directly defined b y
the above construction, Petersen hinted that it can be defined by a limiting
MfM 46 :2
22 7
Figure 1
process or by first adding a point outside the line and afterwards subtractin g
it . In this way one can define k . a, where a is a point in the half plane an d
k E N by first setting a + a = 2a, 2a + a = 3a etc: "in the usual way we
can extend to fractional and irrational values of k" [Petersen 1885, 4] . Now
Petersen introduced two arbitrary but fixed half lines OP and OR and chose
a point called 1 and i on them . Every point of the half plane can now b e
decomposed as a sum of a point on OR and a point on OP and therefore
has the form a • 1 + b . i, a, b E R or as Petersen wrote a + bi . He did no t
remark that he had here confused the real number 1 and the point 1 o n
OR. He remarked that if one chooses L at infinity, OR perpendicular t o
OP and Oi = 01 then one would get the usual description of the complex
numbers . That this usual description had been presented by Wessel to th e
same learned society to whom Petersen addressed his talk was unknown t o
him as well as to all his contemporaries .
Finally Petersen introduced multiplication . He remarked that in order t o
"obtain the usual laws of calculation" one must assume tha t
(a + ib)(a l + ibi ) = aa l + (ab i + bal )i + bb 1 i2
(1)
(distributativity . associativity and commutativity are used, but Petersen di d
not specify which "laws of calculation" he specifically referred to) . Moreover, in order for this to make sense (for the "group" to be closed unde r
multiplication) i2 must be of the form a + iß, a, E R . "It is most nat ural to put i2 = -1" he continued without explaining in what sense thi s
choice was the most natural ; nor did he give a geometric explanation o f
the resulting multiplication . Instead he argued that just as lines through
O are "closed subgroups" under addition and subtraction, so the ellips e
228
J. Liitzen
MfM 46 : 2
{x = a + ib a2 + b2 = 1} is the only infinite "subgroup" with respect t o
multiplication and division . How he could miss the positive real axis, i .e.
the half line OP (with O left out), is odd.
3 .2
Ternions
Having hinted at the new possibilities opened by such a more general treat ment of complex numbers Petersen turned to "ternions" . From the outset
he wrote a ternion of the form x + ay + /3z, where 1, a, O are points on th e
three `"axes" (probably orthogonal) and x, y, z E R. Addition is defined b y
the parallelogram rule (vector addition) . In order to define a multiplicatio n
that satisfies the "usual rules" (only commutativity is mentioned explicitly )
we must define a2 , aß and ,(32 , i.e. we must choose ai , bi , ci, (i = 1,2,3), so
that
a2 = aia + b1 ß + c1 ,
n3 = a2a+b20 +c2,
û2 = a3a+b3ß+c3 .
(2)
In fact one should also define 1 2 , in and 10 where 1 is the point chosen o n
the first axis . However, since Petersen confused this with the real number 1
he implicitly assumed that 1 2 = 1, in = a and 10 = ,3 .
Petersen remarked that one cannot chose ai , bi , ci arbitrarily ; it is require d
that expressions of the form am•/3n always give the same result independently
of the order . He did not explicitly state if there were other requirement s
that had to be fulfilled, nor did he spell his one condition out as an explicit
condition in ai , b i and c i because it "is rather complex" . However, he state d
that the requirement could be satisfied in "different ways" and went on t o
consider "one of the simplest", namely :
a 2 =ß~ ß2 = a ,
a~3
=1 .
(3)
As in the two-dimensional case Petersen then sought "groups" that ar e
closed under multiplication and division. He found the surfac e
x3 + y3 + z3 - 3xyz = 1
(4)
as well as the "null surface"
x+y+z=0
(5)
MfM 46:2
22 9
and the "null line"
x
= y= z .
(6)
The null surface and the null line have the additional property that a produc t
lies in the null surface (null line) when one of the factors lies in the nul l
surface (null line) . This means that the product of a point in the null surfac e
and a point on the null line is zero . Thus, there exist zero divisors, an d
division is not always uniquely defined . The null line and the null surfac e
span the whole space (they are orthogonal in this case) and so any ternio n
can be decomposed as a sum of a point .e on the null line and a point p o n
the null plane . Expressed in this way multiplication can be expressed in th e
simple form :
(Q+p)(f'+p')
= ee ' +
pp
'.
Here £ and L' multiply as real numbers and Petersen showed with som e
geometric considerations that points in the null plane multiply like complex
numbers .
Finally he argued (§12) that an n'th degree ternion equation has less than n 2
solutions . However, that is not correct . Indeed, if the polynomial equatio n
bx +
. +hxm = 0
(7)
has coefficients that all belong to the null plane (or all belong to the nul l
line) then all the infinitely many points on the null line (null plane) wil l
solve the equation . More generally, if all the coefficients of
a+bx+
•+hxm =0
(8)
are products of the same null divisor k:
a=ka', b=kb' , . . . ,h=kh'
(9)
then if a', b', . . . , h' are chosen such that
a' +b'x+ . . .+h'xn= £
(10 )
has solutions for infinitely many values of .e for which k - E = 0, then all thes e
infinitely many solutions will solve the original equation . This consequenc e
230
J. Lützen
MfM 46: 2
of the existence of null divisors played a major role in Weierstrass's discussion of hypercomplex numbers (see below), but it was entirely overlooke d
by Petersen .
In the last section of his paper Petersen discussed quaternions, and from the
start he chose Hamilton's quaternions . Of course he mentioned the lack of
commutativity and he mentioned the uniqueness of (right and left) divisio n
by non-zero quaternions . Otherwise he did not explicitly stress that in th e
four dimensional case he had made a choice different from the one in thre e
dimensions . He had sacrificed commutativity but avoided null divisors .
3 .3
General Evaluation
Petersen's 1885 paper is typical of his production . It is elegant, geometricall y
intuitive, not overburdened with rigour, flawed in some details and without a
single reference. It is in fact unclear whether he knew of Weierstrass's pape r
on hypercomplex numbers from 1884 or Schwarz's additions from the same
and the following year . Dedekind's paper of 1885 may not have appeare d
when Petersen gave his talk . If Petersen did not know of Weierstrass's pape r
one must grant him that particularly the section on ternions reveals som e
original insights . However, even then it is characteristic that Petersen di d
not dig deeper . For example he did not ask if there was anything special
about the choices he made in the two, three and four dimensional cases .
The answer is that the choice of i 2 = -1 in the two-dimensional case an d
Hamilton's relations between i, j and k are very special giving the comple x
numbers, and Hamilton ' s quaternions specially nice algebraic properties .
Petersen's choice : a 2 = ß, ß2 = a,
= 1 in the ternion case on the othe r
hand, gives no special results that one cannot obtain with all "suitable "
other choices . It is characteristic of Petersen that he did not make thi s
discovery, and that he did not initially investigate if his arguments could b e
generalized to higher dimensional commutative n-ions .
Fischer-Benzon also translated this paper by Petersen and sent it in 188 6
to Klein for publication in Mathematische Annalen, but this time Klei n
refused . His reason was probably that Petersen's paper could at best be seen
as a simple exemplification of some of the results in Weierstrass's paper i n
Göttinger Nachrichten of 1884.
MfM 46 :2
23 1
4 The Background to Weierstrass's 1883-8 4
Paper
According to his own testimony [Weierstrass 1884, 396] Karl Weierstras s
(1815-1897) lectured on hypercomplex numbers as early as the winte r
semester 1861-62, 1 but his results were not published until 1884 on the
initiative of Hermann Amandus Schwarz (1843-1921) . Schwarz had hear d
Weierstrass's lectures on hypercomplex numbers and had later tried (prob ably in 1877) to reconstruct their contents . However, he got stuck at a particular detail in the argument : Weierstrass had assumed that for a domai n
of n-dimensional hypercomplex numbers there exists an element g such tha t
gn-1
e g g2 g 3
spans the space where e is the neutral element unde r
multiplication . Now Schwarz was unable to prove the theorem that ther e
exists such an element g in each domain of hypercomplex numbers . 2 He
turned to Weierstrass, but as he recalled in 1883 :
I have already once talked with you (Weierstrass) about this point ,
but you had just mentioned other things that were in the foreground
of your interests, and so we did not discuss this question in detail . 3
However, Schwarz did not doubt the truth of the theorem until 1883 when he
visited Paris and met the Greek mathematician Kyparissos Stephanos, wh o
gave him a simple example of the three-dimensional system of hypercomple x
numbers, where the theorem fails . Stephanos's numbers were of the for m
x = aoeo + a l e], + a2e2 , cai E R,
(11 )
where eo, e l , e2 are the three basic units of which eo is the neutral elemen t
under multiplication and the other elements multiply according to the simple rules :
(12 )
etel = 0, ele2 = 0, e2 e2 = O .
have not been able to locate any lecture notes from this lecture which Weierstrass
had to interrupt halfway through the term due to a breakdown .
2 Schwarz's attempts at reconstructing Weierstrass's theory on the basis of lecture note s
of 1863/64 and 1865/66 are preserved in [Ms 1] dated May 15th 1877 . Schwarz proved
the existence of a unit element, but when he arrived at the problem of the existence of g
his notes degenerated into a series of seemingly unfruitful examples .
3 "Ich habe schon einmal mit Ihnen über diesen Punkt gesprochen, aber Sie hatte n
11
gerade andere Gegenstände, welche im Vordergrunde Ihrer Interesse standen, angereg t
und so unterblieb ein genaueres Eingehen auf diesen speciellen Fragepunkt" . [Schwarz t o
Weierstrass
May 9,
1883] .
232
J. Liitzen
MfM 46:2
In this case
x2 = 2ao(c oeo + a l e' + a 2 e2) - aôeo
(13 )
x2 - 2aox + aoeo = 0,
(14)
so that
which shows that eo, x, x 2 are linearly dependent, so that they do not span
the space, no matter how x is chosen .4
When Schwarz returned from Paris he wrote a letter to Weierstrass (fro m
which I have drawn the above information) informing him about Stephanos' s
counterexample and concluded :
Now arises the question : which theorem should replace it? Do th e
consequences that you have drawn from the above theorem e .g. "in a
number domain with an uneven number of units there exists no numbe r
whose square is equal to -e0" still hold true or are they subject to a
modification?
It would really be rather horrible if one had to add the following qualification to all those theorems . "There are cases where this or that
holds true ; however it does not always occur" .
Which modification applies to Hazzidakis's theorem? 5 [Schwarz to
Weierstrass, May 9 1883]
4Kyparissos (or Cyparissos) Stephanos (1857-1917) studied mathematics in Paris until
1884 when he returned to Athens to become a professor at the University and the Technical
Highschool . While in Paris he communicated his counterexample to Schwarz on a shee t
of paper [Ms 2] preserved in the Schwarz Nachlass . Later (in 1888) he published an 1 1
page paper in Greek on "Systems of complex numbers with several symbolic units" i n
the jubilee publication celebrating the 50th anniversary of the University of Athens, an d
at the first International Congress of Mathematicians he gave a talk "Sur les systèmes
associatifs de nombres symboliques" (see the abstract in the "Verhandlungen des erste n
internationalen Mathematiker-Kongresses", Leipzig 1898 pp . 141-142 .) In this talk h e
reduced the study of hypercomplex numbers to the study of trilinear forms . Many of his
other publications, most in French journals, concern binary and ternary forms .
'Johannes Hazzidakis (or Hatzidakis) (1844-1921) studied mathematics in German y
from 1870 to 1873 . Afterwards he was first a professor at the Military School in Athen s
and then in 1884 at the University of this town, at the same time as his compatrio t
Stephanos. He published several papers in Crelle's Journal, but apparently nothing on
hypercomplex numbers . His role in the story is therefore not quite clear (see however th e
end of §4 .1 below . )
MfM 46 :2
233
Weierstrass answered Schwarz in a long letter that he began on May 19t h
and finished on June 27th 6 1883, a letter that Schwarz subsequently read t o
the Königliche Gesellschaft der Wissenschaften zu Göttingen on December
1st of the same year and published the following year in the Göttinge r
Nachrichten, [Weierstrass 1884] .
The short version of Weierstrass's answer runs as follows . He had early
on been aware of the problem, not as a result of the example given b y
Stephanos, but as a result of an analysis of the noncommutative case, wher e
the quaternions give a similar counterexample . However, he had not explicitly called attention to this circumstance in his lectures prior to 1882-8 3
except for showing that it did not occur "in general" . 7 In his 1882-83 lectures he had more explicitly stated the condition that had to be fulfilled t o
avoid the problem, but he had not had the time to explain why this condition was necessary, in Weierstrass's opinion, in order to have a domain o f
hypercomplex numbers that is algebraically interesting at all .
This direct answer to Schwarz's question concludes Weierstrass's letter t o
Schwarz [Weierstrass 1884, 411-414] . In the first main part of the lette r
Weierstrass aimed at giving
. an authentic account of my theory of the complex quantities ; the
notes, even by the best in my audience contain many misunderstand 6 In the published version [Weierstrass 1884] the date is given as "19 .-27 . June" . That
does not correspond to the dates given in the original letter .
7 In his lectures of 1863/64 (only two years after his first treatment of hypercomple x
numbers) Weierstrass does not seem to have mentioned the problem at all . At least in the
lecture note [Ms 3] the tricky point is passed over in silence . Having remarked that fo r
any hypercomplex number x, the numbers e, x, x2 , . . . , x P are linearly dependent (cf. (30 )
below) when p is the dimension (called n below) he simply stated : "Wenn ich irgend eine
Zahl x als Einheit einführe und x mit der Potenze ; Jede Zahl auf die Form zu bringen :
p-1
a= ae° + ale l + a2e2 + . . . aP - I e
"
In English translation : "When I introduce an arbitrary number x as unit and x as it s
powers ; To represent every number on the form
a = eke ° + ale l + a2e 2 + . . . aP-lep-1
it
To be sure this passage is somewhat obscured by its odd grammatical structure and b y
the change of notation (x is called e in the formula), but it is obvious that the studen t
who took the notes did not detect any problem concerning the existence of x (or e or g
in Weierstrass's later notation) . The lecture notes break off only half a page after thi s
introduction of e .
234
J. Liitzen
MfM 46: 2
ings - with or without the fault of the docent - namely in place s
that go beoynd mere reproduction of calculations . 8
This account was devoted to showing that a strong structure theorem abou t
hypercomplex numbers will hold in general, and to show that in the special cases where it does not hold, non trivial algebraic equations can hav e
infinitely many solutions, which made such algebras too unlike the real num bers to be of any importance .
4 .1
Weierstrass's Theory of Hypercomplex Numbers
I shall summarize Weierstrass's argument at some length, not only becaus e
it is of interest in itself, but also because it sets the mathematical an d
historical background for both Petersen's and Dedekind's later papers .
Weierstrass considered a space (he called it a "Gebiet" (domain)) of ndimensional hypercomplex quantities of the for m
6. e 1 + -2e2 + . .+~ne n
(15 )
where ei , . . . ,G, E R and e l , e 2 , . . . e n are n independent basic units . I
shall call this space
He explicitly stated that the space should be closed
under addition, subtraction and multiplication as well as under division i f
it is defined (see below), and he explicitly set down the following axioms :
C.
a+b = b+ a
(a+b)+c
(a+c)+b
(a-b)+b = a
(16)
a• b
(ab) c
a(b+ c)
b
(17 )
(remark the strange mix of the associative and the commutative laws .)
Weierstrass then claimed that it followed from the first set of axioms tha t
addition and subtraction are performed componentwise, i .e .
n
n
if a = ~ a2 ei and b =
(18 )
i
Ø=1
n
then a ± b = ~( a2 ± ß2)e2 .
(19)
i= 1
8 A sad view from a mathematician, many of whose ideas became known through hi s
students!
MfM 46 :2
23 5
Here he has implicitly used that the ei 's themselves can be considered as
hypercomplex numbers and that the "+" in the expressions (18) for a an d
b is the same as the composition used to add two hypercomplex numbers .
Moreover he has assumed that the "multiplication" with the real number s
ai satisfies the distributive law
(ai + ßi) e i .
a i ei + ßiei =
(20 )
Weierstrass did not explicitly define a multiplication of a general hypercomplex number with a real number, but if we do so in the obvious way we can
consider C, , as a real vector space . I shall therefore somewhat anachronisti cally use vector space terminology in the following discussion .
Implicitly using that aj e j ßk ek = aj/3k(ejek) Weierstrass further conclude d
that if a and b are given as in (18) their product is given b y
a 'b
= ~J
(21)
ajA( ej e k) •
Therefore the multiplication is completely determined once ej e k has bee n
determined for all j, k = 1 , 2, . . . , n. If Eijk denote the coordinates of e j e k
in the basis {ei }, i .e. if
n
ej ek =
(22 )
E ijk e i
i= 1
then the n3 real numbers eijk completely determine the multiplication .
The central exercise is to find the conditions that the e ijk 's must satisfy
in order to define acceptable multiplications . First Weierstrass pointed out
that the commutative and "associative" laws (17) will determine a numbe r
of constraining equations . These equations, namel y
First
constraining
requirement
n
eijk
Eijk,Ekrs
k=1
= e ikj
n
J Eisk E krj,
i, j, h = 1, 2, . . . n
(23 )
2, r, 8 = 1, 2, . . •
(24)
n
k=1
were not written out by Weierstrass, but Dedekind [1885, 145] later did .
236
MfM 46:2
J . Lützen
In order to define the quotient b = ~Z l i ei such that b - b = a, the
coordinates 7i must satisfy the equation
n
ai =
(25 )
ijkßj lyk -
j,k = 1
This is possible in a unique way if the determinant
n
e(b)
(26 )
Eijkßj
j
=l
is different from zero. If it is zero for a given b there are either zero o r
infinitely many solutions -y l , . . . , 7n to (25), depending on the value of a.
Now Weierstrass claimed that for n > 2 one cannot define a unique divisio n
for every b O. Indeed this is a consequence of the following analysis .
However, he required that a unique division can be defined for some value s
of b. This amounts to requiring that e(b), which is a homogeneous function o f
degree n in the ,ß's, is not identically zero . This is his second constrainin g
requirement on the numbers Ei?k .
If both requirements are satisfied, Weierstrass claimed that there exists a
unit element eo E Cn such that eo . a = a for all a E C . Schwarz included
his 1877 proof (see footnote 2) of this fact in an appendix.
Now a third constraining requirement was made to answer the proble m
Schwarz had struggled with : Let
x
= Sl e l
+ S2 e2 + - . . +
Sn e re
(27 )
denote an arbitrary element of Cn . Then its powers can be expressed in th e
form
xU =
~
ly) e l + ~ 2L ~ e2 +
(28)
Weierstrass made the third constraining requirement that the determinant
Xo
=
e(P)
µ, v= 1, 2, . . . ,n
(29 )
is not identically zero . He claimed, and Schwarz proved in an appendix, that
this would indeed be the case except for "special values of the system eijk" .
MfM 46:2
23 7
Under this assumption we can therefore choose a quantity g = elel + 2 e 2 +
+nen for which Xo 0 . Then eo, g, g 2 , . . . , g n -1 will be a new basis o f
Cn as Schwarz had wanted .
Moreover, since eo, 9 1i . . . , g n will obviously be linearly dependent g must
satisfy a polynomial equation of the for m
f (g)
=
gn
+ e l g (n-1) + e2g (n-2) + . . . + e o eo = 0.
(30 )
Weierstrass then made the fourth and last constraining requirement on
the e ijk 's, namely that this equation, considered as an equation in C, ha s
n different solutions . That amounts to saying that its discriminant D 2 is
different from zero.
Now the field is laid out for the proof of the main structure theorem : First
g n- i
Weierstrass remarked that multiplying in the basis eo, go g 1 g2
corresponds to multiplying polynomials modulo f (x) where f is the polynomial in (30) . More precisely : to a hypercomplex number
-1
a
=
aig2
(31 )
aix .
(32 )
i=o
we associate the unique polynomia l
n- 1
(Pa
=
i =o
Then
(33 )
a-b=c
if and only if
~Pa ' Pb =
(Pc
mod f.
(34)
Moreover, the classical result of decomposition in unit fractions states tha t
if co is a polynomial of degree less than n then ~(x)
can be decompose d
f(x)
uniquely in the form
(p ( x )
f(x)
__
cP1(x)
tP2(x )
f1( x) + f2(x)
+
. . .
<Pr( x )
fr. ( x )
(35 )
'
238
J. Lützen
MfM 46: 2
where f 1, f2 , . . . , fr are the real irreducible linear or quadratic factors of f
(that by the fourth requirement are all diferent) and where pl, P2, . . . , (p i. are
real numbers or first degree polynomials (of degree less than their respectiv e
denominators) .
That means that if fl , is of the first degree, the subspace GI,, of hypercomplex
numbers associated (by the association (31) (32)) with polynomials o f
the form
f( x )
(x)
Sop (x )
(36 )
is one-dimensional and spanned by the hypercomplex numbers associate d
with f(x). On the other hand if f, , is a quadratic polynomial the subspac e
fu( x )
G 1 of hypercomplex numbers associated with polynomials of the above for m
(36) is two-dimensional and spanned by the hypercomplex number associ ated with w(()) and f (x)) s .
y
Since any hypercomplex number is associated with a polynomial cp of degree
less than n and since cp(x) =
• f (x), equation (35) shows that any
f(x)
hypercomplex number can be written uniquely as a sum of elements in
G 1 , G2 , . . . , Gr . In modern language Cn = G1 G G2
Gr .
Elementary calculations with polynomials modulo f now show that if a E G i
and a' E Gj then
•
a•a' =0
ifi~j,
(37 )
but
a•a'
0
ifi=j
unless a=0 ora ' =0.
(38 )
Assume for the moment that all the subspaces G ; are one-dimensional an d
write eo = g' + g" +
+ g(n) where g (i) E Gi . Then it is easy to see tha t
(ag (P) ) (Og(u)
)
a
_ ( O) g(A )
(39 )
That means that if we use g (i) . . . , g(n) as a new basis for Cn multiplication
is simply reduced to componentwise multiplication :
n
n
A
cxig(i) .
i=1
i=1
n
(mo i 3 )g(i) .
g(i )
i=1
(40 )
MfM 46:2
23 9
With a little more effort Weierstrass was able to prove that it is possible t o
pick bases g (1) , k (u) in the two-dimensional subspace Gµ such that multiplication of two elements of G m, is done as follows :
(ag(µ) + å )(O9 (µ) + ß'k(12)) = (aß - a'ß')9(µ) + (a'ß + aß')k(u)
(41)
i .e. they multiply exactly as the two complex numbers a+ia' and ß+ . We
can therefore think of the two-dimensional subspaces Gi as one-dimensional
complex vector spaces .
Thus Weierstrass had proved the
Main Structure Theorem . If {eijk } satisfies the four constraining requirements one can decompose Cn as a direct sum of one or two dimensiona l
subspaces :
Cn
= Gi + G2 Eb ED Gr
(42)
such that all algebraic computations involving addition, subtraction, multiplication and division are done componentwise, and such that one compute s
with the components in each Gi as with real numbers (in the one-dimensiona l
subspaces) or with complex numbers (in the two-dimensional subspaces) . Or
expressed in a more modern way Cn is isomorphic to a direct sum of copies
of the real and complex numbers with multiplication defined componentwise .
This theorem is a complete and general higher dimensional analogue of th e
theorem found (later) by Petersen for his specific choice of ternions . We
remark that any hypercomplex number with a zero component in at leas t
one of the subspaces Gi will be a zero divisor because it will give zero i f
multiplied with g (i) . Thus as a special corollary Weierstrass has establishe d
that if the dimension of Cn is higher than two (so that there will be at leas t
two terms Gi in the sum BG i ) there must necessarily be zero divisors . 9
This theorem, showing that there do not exist hypercomplex numberfield s
of dimension higher than 2, is often attributed to Weierstrass (see e .g . [Kline
1972, 793]) . It is however worth mentioning that Hermann Hankel (18391873) had already published a proof of the theorem in his book Theori e
9 Strictly speaking the above argument only leads to this conclusion if {6k} satisfies
all the four requirements . However, he also showed that if the last requirement is no t
fulfilled, there is a A such that x A = 0 has infinitely many solutions . This also gives zer o
divisors .
240
J . Lützen
MfM 46: 2
der Complexen Zahlensysteme [1867] . Yet Weierstrass did not find it wort h
mentioning . Did he not know Hankel's book or did he not feel obliged t o
mention it in a "letter" to a friend and student, since he had derived the
theorem long before the appearance of Hankel's book, or did he even feel tha t
Hankel had in some way stolen some of the ideas from his earlier lectures ?
I cannot say on the basis of the available documents . However, it is equall y
conspicuous that Hankel did not refer to Weierstrass's lectures in his book
despite the fact that he was in Berlin and followed courses by Weierstrass in
exactly the year 1861 [Crowe 1972], in which the latter gave his first accoun t
of hypercomplex numbers .
Another question of priority concerns Hazzidakis's contribution . Having
proved that one can choose a basis in the two-dimensional subspaces G i
such that multiplication corresponds to complex multiplication of the components, Weierstrass added in a footnote :
As is well known to you Mr Hazzidakis has arrived at the same resul t
in a different way. [Weierstrass 1884, 406]
It is not quite clear how much of the Main Structure Theorem Hazzidaki s
had proved : Had he proved the whole theorem? This seems unlikely considering the little mention he got . It seems more likely that he had heard of
Weierstrass's decomposition On = ®Gi , and had found the special basis in
the two-dimensional subspaces Gi , in which the multiplication reveals itsel f
as complex multiplication . The way Schwarz referred to Hazzidakis's theorem in his letter to Weierstrass (see §3) even suggests, that Hazzidakis wa s
the first to see that the two-dimensional subspaces Gi behave like complex
numbers .
Weierstrass's first two requirements seem natural, but the last two may see m
artificial . However, Weierstrass showed that if one of them is not satisfied ,
then a polynomial equatio n
a +bx + cx2 + . . . + hxm
=0
(43 )
can have infinitely many solutions even when its coefficients a, b, . . . ,h are
not multiples of the same zero divisor . On the other hand he could show
that if the requirements are satisfied, a polynomial equation has finitel y
many solutions except when its coefficients a, b, c, . . . ,h are multiples of the
same zero divisors. Since the existence of zero divisors cannot be avoide d
(for n > 3) this is the closest one can get to the usual fundamental theorem
MfM 46:2
24 1
of algebra . Conditions 3 and 4 therefore ensure that we keep as close a s
possible to ordinary algebra, and they are therefore natural according t o
Weierstrass .
5 Petersen's 1887 Pape r
In 1887 Petersen published his own treatment of general n-dimensional hypercomplex numbers in Göttinger Nachrichten . It is phrased as an alter native to Weierstrass's treatment, of which he had become aware in th e
meantime (probably through Klein or Fischer-Benzon) . In fact, much of the
treatment follows Weierstrass closely, but there are notable differences . First
of all Petersen used complex numbers . Initially he defined hypercomple x
numbers as quantities of the form ] . e l + 2e2 + • n e n where i E R. Her e
he followed Weierstrass . However, when he had defined the quantity g and
the polynomial f (30), he used all its real and complex roots x i, x2, . . . ,x,,,
to define a new set of elements : pi , p2 , . . . , pn by:
pi =
f(x
.Î'(x i)( x) xz)' (x = g)'
(44 )
where it is understood that one must set x = g after having made the
division . Petersen now showed that the new set of points p i multiplies each
other in the following simple wa y
pi - Pi = Si?pi
where 6i, ß
1
= 0
for
for
i = j,
i L j.
(45)
The p-system is also a basis and (42) implies that expressed in this sys tem, multiplication is simply termwise multiplication of coordinates, i .e. if
a l , a2 i . . . , an are a's coordinates and b 1 , b 2 , . . . , bn are b's coordinates in
the p-basis, then a - b has the coordinates a l bs, a 2 b2 , . . . , an b n . In the same
coordinates the Null set has the equatio n
x l x 2 . . .x n = 0
(46 )
and consists of the n (n - 1)-dimensional subspaces :
xi=0,x2=0, . . .,xn=0 .
(47)
242
J. Liitzen
MfM 46 : 2
Petersen then proceeded with the example which he had given in his 188 5
paper . However, this time he explicitly mentioned the multiplication law s
for the first "basis vector" el (which he had implicitly identified with 1
earlier) and he decomposed the null-surface from 1885 into two comple x
null-lines . He also generalized his search for "subgroups" invariant unde r
multiplication and concluded that they are all given by an equation of th e
form
x a 1 x~2 . . .
1
2
x an
=
1,~
(48 )
except the earlier mentioned null-spaces and their intersections . Petersen
then went on to analyse Weierstrass's four requirements and first showe d
that they were independent in the sense that the fulfilment of three of the m
does not ensure the fulfilment of the last one . He finally found a "surprisingly
simple" requirement that he proved to be equivalent to Weierstrass's las t
three requirements : Consider the equations of condition (22 )
ej ek
= L ez.jke z
(49 )
2- 1
as an equation in n numbers (complex numbers) el , e 2 , . . . , en . Then Weierstrass's three conditions are satisfied if and only if one can find n sets o f
solutions
etil e z2 ) eå3) . . . ,
e in)
i = 1, 2, . . . , n,
(50 )
such that the determinant 1 % is different from zero . Expressed in vector space language Petersen ' s argument runs somewhat along the followin g
lines.
First assume Weierstrass's conditions are satisfied . Then the previous argument shows that there exists a basis , P 2 i, .. . . , pn in which multiplication is
simply coordinatewise multiplication . If we express the original basis vector s
el , e 2 , . . . , e, in this system
n
\\
ej =
(51 )
e2J pz ,
i=i
then the original equations of condition separate according to the coordinates into the equations :
e ij eik
=
efj k ez.e .
(52)
MfM 46 :2
Hypercomplez Numbers in the 1880es
243
Thus ei', ei2, . . . , ein is a set of solutions to (49) for all i = 1, 2, . . . ,n and
since {ei } is a basis the determinant ei,j 1 of its coordinate vectors is non
zero.
Conversely, assume we can find solutions eil, eil, . . . , ein of (49), i =
1, 2, . . . ,n so that Ieij l
O . That means that if we consider the n vector s
in Rn
ei l
ei =
eil
i = 1, 2, . . . ,n
(53 )
ein
they are a basis of C n and under coordinatewise multiplication they satisf y
the equations of condition (49) . That means that the product determined
on Cn by the equation of condition (49) has the property that it can b e
described as coordinatewise product in a suitable basis, or said differently ,
Cn = C ® C ® • ® C, and that can only happen if Weierstrass's condition s
are satisfied . Indeed these conditions are satisfied by the eijk that defin e
coordinatewise multiplication, and they remain true under a change of basis .
So when Petersen in his condition asks for n solutions of the equations (49) ,
he really asks for the construction of a set of linearly independent vector s
el , e2 , . . . , en in Cn , such that multiplication defined coordinatewise in th e
standard base in Cn will yield the equations of condition (49) . Since the
existence of a basis in which multiplication acts termwise is equivalent to
Weierstrass's conditions, Petersen's conditions are clearly correct .
One may question the value of Petersen's result . Indeed the structure o f
Weierstrass's argument is to establish that under rather weak condition s
(1-4) fulfilled by a host of (almost all) choices of 5ijk (and by all choice s
that will yield acceptable algebras at all), one can prove that the multiplicative structure so to speak, separates out in the sense that expressed i n
a suitable basis it is nothing but coordinatewise multiplication . Conversely
the existence of a basis in which the product is expressed coordinatewis e
implies Weierstrass's conditions, but of course, this converse result is les s
interesting or surprising . What Petersen's conditions ask is to construct th e
basis in which the multiplication is termwise in order to verify Weierstrass' s
conditions . It seems to verify the less complicated by the more complicated .
244
J. Lützen
MfM 46:2
6 The Fate of Petersen's Paper
Petersen wrote his paper in German and sent it in May 1887 to Schwarz
for publication in the Göttinger Nachrichten . At this time Petersen knew
Schwarz quite well 10 . They had met for the first time in 1877 in Copenhagen ,
and later the same year Petersen had sent Schwarz a paper on integratio n
of differential equations in closed form for publication in the Nachrichten .
Already at this time Schwarz was faced with the problems involved in publishing Petersen's ideas . He asked Petersen to correct imprecisions and make
references to the relevant literature . Moreover, when the paper appeared i n
1878, Leo Königsberger (1837-1921) wrote a letter to Petersen calling hi s
attention to a paper he had published the previous year in Grelle's Journa l
in which he had proved a more general theorem .
In 1886 (or 87) Petersen visited Schwarz in Göttingen . It may have been
a discussion about hypercomplex numbers on this occasion that made Pe tersen compose his paper and send it to Schwarz . Schwarz immediately presented it to the Königliche Gesellschaft der Wissenschaften zu Göttingen on
May 1887, but already five days later, while preparing the manuscript fo r
publication, he found himself in a situation similar to the one he had been i n
ten years earlier : To be sure Petersen had referred explicitly to Weierstrass' s
paper and to a paper by Schwarz himself in which he proved that the sub spaces G i do not depend on the choice of the quantity g used in its construction [Schwarz 1884] . He had also referred to a paper by Otto Hölder (1859 1937) [1886], in which Hölder showed that the subspaces G i are uniquel y
determined by the requirements that a) they are one- or two-dimensional ,
b) every point in Cn can be uniquely written as a sum g i + g2 + • • • + gr
where gi c G i , c) that gi • gj = 0 if gi E Gi , gj E
j and d) gi • gz E Gi
if gi , gZ E Gi and the product is only 0 if gi or g'i are zero.
However, as Schwarz informed Petersen, he had missed a central paper by
Richard Dedekind (1831-1916) from the intervening year 1885 :
. . . you have not quoted a paper by Professor Dedekind in Braunschweig, that was printed in the volume of 1885 of our Nachrichte n
on page 141-159, and to which Mr Dedekind has also published a
supplement in the present volume (1887 page 1-7) . The theorem that
"The following information is drawn from the Petersen-Schwarz correspondenc e
[Petersen-Schwarz 1877-87] .
MfM 46:2
24 5
you have discovered is already to be found by Mr Dedekind . Therefore
it is not enough just to add a reference to Mr Dedekind's paper to you r
list of references . Rather, it is absolutely necessary that you expres s
yourself clearly about the relation between the content of your pape r
and the content of Dedekind's two articles . You alone can do that .
Therefore I hurry to ask you to acquaint yourself intensively with th e
papers by Dedekind, so that you can complete your paper and you r
quotes . [Schwarz to Petersen, May 13, 1887]
The same month Dedekind himself also wrote Petersen a letter explaining hi s
approach. Apparently Schwarz had informed Dedekind about the content o f
Petersen's paper, and it may very well be Schwarz who had suggested tha t
Dedekind wrote directly to Petersen . Schwarz's letter shows clearly that h e
knew it would be difficult to make Petersen read Dedekind's paper and tak e
it into account in a serious way. This proved to be correct . Petersen jus t
added the following reference to Dedekind :
The main theorem that I intend to announce in this paper says tha t
each multiplication rule for complex numbers composed of n basic
units, that are allowed according to the basic conditions stated b y
Mr Weierstrass, is determined by n systems of each n real or ordinary complex quantities such that the determinant formed by thes e
n2 quantities has a value different from zero . This main theorem has
already been formulated by Mr Dedekind . The proof I have found
seems to me to be simpler than the one published by Mr Dedekind .
[Petersen 1887, 489-490] .
Let us turn to Dedekind's paper and compare it to Petersen's . One similarity
between them which opposes them to Weierstrass is that they both treat
the problem in a complex setting. Here Dedekind is more consistent than
Petersen in dealing from the start with complex coefficients (we would sa y
he deals with vector spaces over (C) whereas Petersen first considered rea l
coefficients, and only later introduced complex numbers . Dedekind's main
theorem says (in slightly modernized version) :
If {eij } is a matrix with non zero determinant and we define eIJk such tha t
its column vectors ei multiplied coordinatewise will satisfy the equation s
(20,49)
eres
= i , e tiEi,rs
r,s= 1,2, . . .,n
(54)
246
J. Lützen
MfM 46:2
then the e21.,s will satisfy the commutativity and associativity conditions (23 )
and (24) and the condition that the determinant o f
0=
{>
cïiirs> r,s
(with o-i =
(55 )
is non zero . Conversely, and that is the main part of the theorem, if eirs
satisfy these conditions they arise in the described way from a matrix {eij}
with non zero determinant .
Thus Dedekind's theorem did not directly establish the relation to Weierstrass's conditions but to another condition on the determinant 0 that i s
definitely simpler to check . So, when Dedekind in the introduction to hi s
paper spoke of "a small simplification of the permissibility conditions formulated by Weierstrass", he did not talk about Petersen's condition but abou t
the non vanishing of ßo1 . Although it is a direct consequence of Dedekind' s
argument, he did not specify that one may think of the numbers ei3 in each
row as a set of solutions to the basic equations (54) . To make this explicit
is Petersen's main contribution .
It is also arguably correct that Petersen's proof, as he wrote himself, is simpler than Dedekind's . However, that is mostly due to the fact that Dedekind
proved a somewhat deeper theorem . Indeed, Weierstrass's conditions are s o
to speak proof-generated to fit the proof that a set of basic equations (54)
with coefficients eijk satisfying these conditions, will yield a basis in which
multiplication is coordinatewise, leading in turn to the matrix {%} . To
deduce the same conclusion from Dedekind's one, seemingly simpler, requirement is a long story.
On September 9, 1887 four months after presenting Petersen's paper t o
the Göttingen Academy, Schwarz was again busy preparing Petersen' s
paper for publication, and he discovered that another major correctio n
was needed : Petersen had accused Weierstrass of an imprecision (Ungenauigkeit), and had offered a counterexample to Weierstrass's proof. It seem s
from Schwartz ' s letter to Petersen (September 9, 1887) that Petersen suggested that by taking g = eo in his favourite example of ternions 02) ,
one would not get the desired result even though Xo is not identically zer o
for this system. However, as Schwarz pointed out to Petersen, Xo is equa l
to zero for this particular choice of g (or rather of (1,2>3) = (1,0,0)) .
He further argued that Weierstrass had rather explicitly stated that one
MfM 46 :2
24 7
must pick g in such a way that Xo is different from zero for this particula r
g . Therefore he suggested to Petersen that "either your remark about th e
imprecision must be deleted completely or it must be phrased quite differently" (Schwarz to Petersen, September 9, 1887) . Petersen chose to erase
any criticism of Weierstrass .
Later the same day Schwarz sent Petersen a new letter suggesting a simplifi cation of an example in Petersen's paper . Petersen had proved that the three
quantities e, X0 , D (see (26) and (29)) were related through the equation :
Xo = eD .
(56 )
He had therefore asked the question : is Weierstrass's fourth condition D 0
not a consequence of the second (e 0) and the third (X 0
0)? However ,
he correctly observed that the equality (56) had been derived under the condition that X0 , e and D are all non zero, and that therefore one cannot mak e
the suggested conclusion. He had even given the following two-dimensiona l
example
e1 = e2, ele2 = -el - ae2, e2 = ael + (a 2 - 1)e2,
(57 )
which for a = 2 gives D equal to zero while X0 and e are not identically
zero . He had learned this example from his friend Frederik Bing (1839 1912) . However, Schwarz remarked that in this case the identity element i s
eo = -ae l - e2 , and he suggested using e0 and el + zaeo as new basic units .
If we let 6 denote 4 (a2 - 4) the fundamental multiplication laws simply loo k
like :
2 = e0
eo
eoei
= el
el2 = beo .
(58)
He even indicated that any multiplication law for two-dimensional number s
could be reduced to this one .
For this reason I cannot at all accept that an underhand gaine is bein g
played in Mr . Bing's example . "
In Schwarz's example e = e - S 2 and X0 = e2( e - 6a) and a general
quantity g = el eo + e2 e 1 satisfies the equation
g2 - 2elg +
11
( i-b
2)e o
= 0,
(59 )
"dass bei dem Beispiele des Herrn Bing thatsächlich Verstecken gespielt wird" .
[Schwarz to Petersen, September 9, 1887] .
248
J. Lützen
MfM 46 : 2
whose discriminant D2 is 8 2 . Therefore f = 0 gives the desired resul t
D2 = 0, X0 0, e 0 . Schwarz even claimed to have known this exampl e
for many years .
Petersen again followed Schwarz's advice, and replaced Bing's example wit h
that of Schwarz, leaving out any reference to either of his two friends . With
these corrections Schwarz considered Petersen's paper "as a real progress "
(Schwarz to Petersen, September 9, 1887) .
The printer sent Petersen a proof on October 4, 1887, but ten days later
Schwarz informed Petersen about a new problem . He had asked Otto Hölder
to go over the proofs, and Hölder had discovered that a remark Peterse n
had made about "the factors which are substitution determinants must als o
be made about r (Schwarz to Petersen, October 19, 1887 [the meaning i s
not entirely clear to me]). However, as Schwarz remarked to Petersen :
Now this change or correction cannot be made in a twinkling, becaus e
there is a risk that it may easily lead to a worsening rather than a n
improvement. [Schwarz to Petersen, October 19, 1887 ]
Therefore Schwarz simply left out the remark . He also announced to Petersen that Hölder would write and explain this point and other points t o
him in private .
Thus, when Petersen's paper finally appeared in the November 16, 1887
issue of Göttinger Nachrichten it bore a heavy imprint of its editor H .A.
Schwarz .
7
Dedekind against Weierstrass, Göttinge n
against Berli n
In addition to presenting his complex alternative to Weierstrass's treatment
and his simpler condition Dedekind had two goals with his 1885 paper on
hypercomplex numbers . He called attention to §159 in his 10th supplement
to his 1871 edition of Dirichlet's number theory . Here Dedekind had considered algebraic field extensions Q(r) of the rationale ("endliche Körper"
in his terminology) . If r is a solution of an irreducible equation of degree n
over Q then elements of Q(r) can be uniquely represented a s
1
+ e2r + . . . + Gr n-1
(60)
MfM 46:2
24 9
where i (i = 1, 2, . . . , n) are rational numbers .
In his 1885 paper Dedekind pointed out that if we allow the coefficients z to
be real (or complex as Dedekind preferred), one would get an expression fo r
a hypercomplex number written in the basis e, g i , g 2 , . . . , g n-1(g = r) wher e
g is a solution of the given equation . He did not mention hypercomplex num bers explicitly in his Dirichlet supplement, but in the 1885 paper he claime d
that he had "dealt with them in passing", implying that this happened i n
1871 . Thus the claim of priority he seems to make is that he was the first t o
publish on something strongly related . Weierstrass could still claim priority for his earlier discussion of hypercomplex numbers . Indeed, in his paper
from 1884 he had written that he had "years ago conceived the followin g
considerations" to which he added in a footnote : "I lectured on this subject
for the first time during the winter semester 1861-62" [Weierstrass 1884,
396] . Dedekind did not question how many of the "considerations" Weierstrass knew about in 1861-62, nor did he push his own claim of priority
beyond the vague hint to his 1871 supplement.
He was much more insistent on another matter, namely the evaluation o f
the following paragraph in a paper by Carl Friedrich Gauss (1777-1855) on
biquadratic reciprocity :
The author has planned to give later a more complete treatment o f
the subject which is really only touched upon in the present paper .
Here the following question will then also find its answer . Why the
relations between things that present a manifold of more than tw o
dimensions cannot yield other types of quantities which are allowed i n
the ordinary arithmetic . [Gauss 1831, Werke II, 178] .
This quote was the point of departure for Weierstrass's investigations of
hypercomplex numbers . He interpreted Gauss as saying that it is impossibl e
to define a multiplication for hypercomplex numbers of dimension 3 or mor e
that satisfies all the arithmetical rules of the real numbers (i .e. that makes
it a field) . Having proved his main theorem Weierstrass concluded :
When I compare the results of the preceding investigations with th e
quoted Gaussian remark, to wit complex quantities with more tha n
two basic units are not allowed in the usual arithmetic, it seems t o
me that Gauss believed this impossibility was founded on the fac t
that the product of two quantities (when n > 2) can vanish without
any of its factors vanishing . For if he had not considered this fact a n
insurmountable hindrance, he would probably not have failed to notice
250
MfM 46: 2
J. Lützen
that it is possible to substantiate an arithmetic of such quantities i n
which the theorems are either identical to those of the arithmetic o f
the usual complex quantities, or at least find their analogue there .
In that case he would undoubtedly have modified his claim to sa y
that although the introduction of the general complex quantities in
arithmetic is not impossible, it is superfluous . In fact, the theore m
stated above (page 407) [Weierstrass's main structure theorem] show s
that the arithmetic of the general complex quantities cannot lead t o
any results that cannot be deduced from the results of the theory of
complex quantities with one or with two basic units. [Weierstrass 1884,
410-11] .
Thus Weierstrass thought that Gauss's insight into the matter was limited t o
seeing that all the usual arithmetic laws, including unique division with no n
zero elements, could not be generalized beyond 2 dimensions . Weierstrass did
not think that Gauss had investigated what would happen if one dropped
the requirement of unique division with non zero elements, and he eve n
suggested how Gauss ought to have phrased himself if he had seen what
Weierstrass later saw .
Such a mild criticism of the Princeps Mathemraticorurn was too much fo r
Gauss's former student Dedekind .
He agreed with the last part of Weierstrass's quote above and argued that
this was indeed the meaning of the quote by Gauss . However, he phrased i t
somewhat differently . In fact, as admitted above, none of the three mathematicians we have considered in this paper used the language of vecto r
spaces, and here is a point where it matters . Instead of talking about expressing the vectors e l , e2, . . . , en, in a base in which multiplication is coordinatewise multiplication, Dedekind considered one system of complex
numbers
el , e2, . . . ;en ,
but considered the system to be multiple valued, being able to obtain the n
values
elr e r2
e
u
n
l
e2
e(n) B(n)
1
2
en
en
(61)
e~n )
MfM 46:2
25 1
Here we would consider the columns to be the coordinates of the vectors ei
in the p-base . Dedekind, however, wrote that any polynomial equation i n
the ei 's would hold true if it hold true whichever of the multiple values o f
the system we substitute in the equation . More particularly, if we expres s
everything in terms of e0, g, g 2 , . . . , g n as above, any equation in the e's can
be translated into an equation in g : w(g) = 0(g), and such an equation
is fulfilled, whichever of the roots x 1 , x 2 , . . . , xn of the polynomial (30) w e
insert in the equation . I.e. cp(r) = qP(r) in this multiple-valued algebra mean s
<p(r)
(r) = 0 mod f,
(62 )
(this is similar to Weierstrass's observation) . Thus the arithmetic of hyper complex numbers is nothing but multiple-valued arithmetic of ordinary com plex numbers . As an example Dedekind mentioned the system of ternion s
determined by the following basic equation :
e21
e 22
e 22
el e3
ege l
e l e2
-
-
- 2e 1 - e 2 - 2e 3
2e 1 - 2e2 - e 3
- e 1 - 2e 2 - 2e 3
el + e 2
e2 + e 3
e1
+ e3
(63)
It is an easy matter to check that this system satisfies Dedekind's basi c
condition . But as pointed out by Dedekind, calculation with these quantitie s
is indistinguishable from calculation with the three-valued period s
el = r + r-1
e2 = r2 + r 2,
e3 = r 2 +r 2
(64)
of the circle division polynomia l
r s +r 5 +r4 +r 3 +r 2
+r+1
=0,
(65 )
when r runs through the different roots of this equation . Knowing Gauss' s
interest in the periods of the circle division polynomials [Gauss 1801] ,
Dedekind suggested that Gauss had already obtained the insight that Weierstrass published, but that he drew the conclusion that such an arithmetic
was not new, but corresponded to an arithmetic of multiple-valued comple x
numbers that had been studied long ago . This new interpretation of Gauss' s
somewhat ambiguous statement gives much more credit to Gauss . Dedekind
252
J. Lützen
MfM 46 : 2
put forward his interpretation already in his 1885 paper, but since he dis covered that he had been misunderstood, he expressed it more explicitly i n
his supplement of 1887 .
This small difference between Weierstrass's and Dedekind's opinion is symptomatic of the tensions between Berlin and Göttingen at the time . It is a
clash between a Göttingen educated former student of Gauss, who revere d
his master, and a Berlin mathematician who did not mind attributing som e
limits to the insights of the Göttingen Princeps .
Acknowledgments
I wish to thank Gert Schubring for many valuable suggestions and refer ences to manuscripts as well as the staff at the Berlin-Brandenburgisch e
Akademie der Wissenschaften for locating the central letter from Schwar z
to Weierstrass .
References
Letters and manuscript s
The Julius Petersen correspondence, including the letters from H .A . Schwarz, is preserve d
in the Royal Library in Copenhagen, Ny Kgl . Saml . 3259, 4°, I (A-N) and II (O-Z) .
The letter of May 9, 1883 from H .A . Schwarz to Weierstrass is preserved in the Nachlass
Schwarz, Nr . 449 and 1254 at the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin .
Ms 1, Attempt to reconstruct Weierstrass's theory of hypercomplex numbers . Unsigned
but apparently in H.A . Schwarz's handwriting . Dated May 15th 1877, 9 pages .
Nachlass Schwarz Nr . 440 at the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin .
Ms 2, One sheet dated May 31st 1883 written in French and signed by Cyparisso s
Stephanos, containing his counterexample mentioned in section 4 above . Nachlas s
Schwarz, Nr . 720 at the Berlin-Brandenburgische Akademie der Wissenschaften ,
Berlin .
Ms 3, Anonymous lecture notes taken from Weierstrass's lectures during the Win ter semester 1863/64 . Nachlass Schwarz, Nr 440 at the Berlin-Brandenburgisch e
Akademie der Wissenschaften, Berlin .
MfM 46 :2
25 3
Published materia l
Christiansen, M . et al., 1992, "Julius Petersen, annotated bibliography", Discrete Mathematics 100, 83-97.
Crowe, M .J ., 1972, "Hankel, Hermann", Biography in Dictionary of Scientific Biograph y
(ed . Gillispie, C .C .), VI, 95-96 .
Dedekind, R., 1885, "Zur Theorie der aus n Haupteinheiten gebildeten complexen
Grössen", Nachrichten Königl. Ges . der Wiss . zu Göttingen, 141-159 .
Dedekind, R ., 1887, " Erläuterungen zur Theorie der sogen . allgemeinen complexen
Grössen", Nachrichten Königl. Ges . der Wiss . zu Göttingen, 1-7 .
Gauss, C .F ., 1801, Disquisitiones Arithmeticae. Braunschweig, Werke 1 .
Gauss, C .F ., 1863, "Theoria residuorum biquadraticorum, Commentatio secunda", Göttingische gelehrte Anzeigen, 1831 . Werke 2, 169-178 .
Hankel, H ., 1867, Theorie der Complexen Zahlensysteme, Leipzig .
Wilder, O ., 1886, "Bemerkung zu der Mittheilung des Herrn Weierstrass : Zur Theorie der
aus n Haupteinheiten gebildeten complexen Grössen", Nachrichten Königl . Des .
der Wiss . zu Göttingen, 241-244 .
Klein, F ., 1871, " Uber die sogenannte Nicht-Euklidische Geometrie", Math. Annalen 4 ,
571-625 .
Kline, M ., 1972, Mathematical Thought from Ancient to Modern Times, New York .
Lützen, J ., Sabidussi, G . and Toft, B ., 1992, "Julius Petersen 1839-1910 . A biography" ,
Discrete Mathematics 100, 9-82 .
Petersen, J ., 1866/79, Methoder og Theorier til Løsning of Geometriske Konstruktionsopgaver, Schonberg, Kjøbenhavn 1866, 2 . enlarged edition 1879 . Translated int o
many languages .
Petersen, J ., 1877, De algebraiske Ligningers Theori, Høst & Søn, Kjøbenhavn 1877 .
German transl . 1878 . French transl . 1897 .
Petersen, J ., 1878, "Beweis eines Lehrsatzes betreffend die Integration algebraischer Differentialausdrücke beziehungsweise algebraischer Differentialgleichungen unter geschlossener Form", Nachrichten Königl. Ges . der Wiss . zu Göttingen, 68-88 .
Petersen, J ., 1881, Forelæsninger over Statik, Høst & Søn, Kjøbenhavn 1881 . German
translation (Lehrbuch der Statik fester Körper), 1882 .
Petersen, J ., 1887, "Om Mathematikens Grundbegreber . Bevis for Sætningen om Trekantens Vinkelsum", Tidsskr . f. Math . (1883) (5) 1, 3-11 . German transl : "Bemerkungen über den Beweis des Satzes von der Winkelsumme des Dreiecks", Math . Annalen 29 (1887), 239-246 .
Petersen, J ., 1884, Forelæsninger over Kinematik, Høst & Søn, Kjøbenhavn 1884 . Germa n
transl . (Kinematik) 1884 .
Petersen, J ., "Om Algebraens Grundbegreber", Tidsskr. f. Math . (1885) (5) 3, 1-22 .
Petersen, J ., 1887a, "Uber n-dimensionale komplexe Zahlen", Nachrichten Königl . Ges .
der Wiss . zu Göttingen, 489-502 .
Petersen, J ., 1887b, Forelæsninger over Dynamik, Høst & Søn, Kjøbenhavn 1887 . Germa n
transl . (Dynamik) 1887 .
Poincaré, H ., 1882, "Théorie des groupes Fuchsiens", Acta Math . 1, 1-62 .
Weierstrass, K ., 1884, "Zur Theorie der aus n Haupteinheiten gebildeten complexe n
Grössen", Nachrichten Königl. Ges . der Wiss . zu Göttingen, 395-410 ; Math . Werke
2,311-332 .
254
J. Lützen
MfM 46 : 2
Wussing, H ., 1984--1969 The Genesis of the Abstract Group Concept, Cambridge Mass .
1984 ; transi . from Die Genesis des abstrakten Gruppenbegriffes, Berlin 1969 .
MfM 46:2
25 5
Priority Claims and Mathematical Values : Disputes
over Quaternions at the end of the ineteent h
Century
Tom Archibald *
1 Introduction : Priority and Values in the Case o f
Wessel
In this volume as well as in general treatises on the history of mathematic s
Wessel is identified as the first to arrive at a satisfactory geometric representation of complex numbers . However, Wessel's idea scarcely exhaust s
the relationship between imaginaries and geometry, as the paper of David
Rowe in this volume shows . The fact, important to Wessel and others, that
his geometric representation gives a way of making imaginary numbers real ,
likewise does not account for our declaring this particular creation as a
"first" which is worthy of interest . For as other papers in this volume indicate, there are earlier efforts at resolving this imaginary character int o
reality, by Wallis, Foncenex, and so on . We therefore have to ask why i t
is this particular thing, the geometric representation of complex numbers ,
that we select for attention .
It seems to me that the reasons for this lie embedded in our mathematical values . This priority question was of key interest in the second half of
the nineteenth century precisely because of the enormous importance of th e
theory of functions of a complex variable, a theory in which this representation is a fundamental tool . The work of Riemann on complex analysis ,
* Dept . of Mathematics and Statistics, Acadia University, Wolfville N . S . BOP 1X0 ,
Canada
256
T. Archibald
MfM 46 :2
and the geometric interpretation of the Cauchy integral by Puiseux, Brio t
and Bouquet showed the basic importance of the complex plane . If we loo k
at the way in which Wessel's priority came to be accepted, we see tha t
the key event appears originally to have been taken as the 1831 Anzeig e
of Gauss, after which the "usual" complex plane appears to have becom e
generally known and accepted, though the details of its reception remain t o
be worked out. In any case, looking at Hermann Hankel's 1867 Theorie der
complexen Zahlensysteme, we get the impression that during Hankel's education the view of Gauss as originator of this representation was generall y
held. Hankel however revised this view, following Cauchy by mentioning a
number of less well-known writers, and giving priority to Argand . In my
view this is because for Hankel the exact form of the Gaussian inventio n
is the key innovation, rather than for example the work of Buée, also mentioned by Cauchy. Thus an earlier formulation - that of Argand - whic h
is essentially identical to that of Gauss counts as a prior discovery, whereas
others less completely identical to Gauss's do not . Hankel's line was followed by Hoüel (1874), thus becoming the general view in France as well .
The discovery of the fact that Wessel had done essentially the same thing a s
Argand thus gave weight to the claim of Valentiner, Thiele and Lie, and i t
is Wessel who generally gets credit from then on, as we can see for exampl e
in the Eneyklopädie der mathematischen Wissenschaften . Thus the importance of the Wessel discovery stems from the importance originally given t o
the Gaussian contribution, via the historical revision to include Argand .
2 The Quaternion Priority Question
Quaternions are a different case . Now more marginal mathematically, thei r
geometric representation presents a more complex set of issues in the resolution of priority. The efforts of Wessel to create an algebra adequate t o
handle rotations algorithmically clearly prefigure quaternions, as does work
of other writers in the same vein, most notably Servois, Argand's contempo rary, in the Annales de Gergonne of 1813 . To grasp the negotiated characte r
of mathematical priority, let us examine a dispute between P . G. Tait an d
Felix Klein which centres precisely on the issue of the extent to which th e
notion of rotation, and geometrical transformation more generally, capture s
the essence of the quaternion idea.
MfM 46:2
Priority Claims and Quaternions
257
On December 18, 1899, Peter Guthrie Tait read one of his last papers be fore the Royal Society of Edinburgh . Tait titled his paper "On the Clai m
recently made for Gauss to the Invention (not the Discovery) of Quater nions" (emphasis and parentheses in the original) . Tait's paper conteste d
the argument advanced by Felix Klein to the effect that the essential ide a
of quaternions had been discovered by Gauss as early as 1820, based on a
manuscript fragment found in the course of preparing the Gauss collecte d
works . Tait's argument rested on two related points : first, that Klein had
misunderstood what a quaternion was; and second, that the discovery o f
the quaternion operators (as opposed to the mere invention of the four dimensional algebra with imaginary units i, j and k) was far more profound
and consequential than anything envisaged by Gauss .
Tait thus was defending the priority of his old mentor Hamilton, as well as
the mathematical ability of the British in general, against an attack from
abroad . Indeed, Tait was no stranger to this genre : his Sketch of Thermodynamics of 1868 had begun a long series of efforts to assert priority for variou s
British colleagues in diverse fields of mathematical and physical endeavour .
There is nothing particularly new about priority disputes in the sciences .
However, the specifically national context of the disputes the effort t o
claim glory not only for an individual but for a nation - took on a particularly strident form in the late nineteenth century . Not surprisingly, thi s
coincides with the upsurge of national communities in mathematics an d
with national and international meetings.
In the absence of plagiarism, disputes about priority arise because of wha t
is usually called independent discovery . Even when such discoveries are no t
synchronous, they force us to think about the historical circumstances unde r
which they occur and the conditions which make such events possible . As
Thomas Kuhn has discussed (Kuhn, 1959) for the case of simultaneou s
discovery, it is often the case that there is relatively little overlap in th e
actual findings, and this is certainly true for our case . As is clear fro m
many of the other papers in this volume, considerable variation is possibl e
in representing directed quantity mathematically. In mathematics, we hav e
come to expect that such representations are to some extent translatabl e
into one another with the benefit of hindsight . However such translations
often overlook important aspects of the author's vision, and the difference s
between theories and their translations offer important fodder for dispute s
between interested actors.
T. Archibald
258
MfM 46:2
3 The Background to Klein's Clai m
3.1 Germany
One important part of the German background to Klein's claim is presente d
in Hankel's (1867) Theorie der complexen Zahlensysteme . Hankel's historical remarks concerning the geometrical representation of complex number s
mention a number of eighteenth-century precursors, but he follows Cauchy
in giving priority to Argand (See Belhoste 1991 p . 232) . Noting one French
and one English rediscovery (by Mourey and Warren respectively) Hankel
then asserts the importance of the German rediscovery by Gauss (1831) ,
which brought this representation
as well as the very term complex num ber
to the attention of a wider mathematical public .
It should be noted that there had already been a Franco-German priorit y
dispute in a closely related matter, namely between Grassmann in Ger many and Cauchy and Barré de Saint-Venant in France . The matter was
raised by Grassmann, and investigated in France by the Académie des Sciences, who sagely appointed a commission to investigate (which include d
Cauchy) . This commission never reported, but the very fact of the disput e
which concerned the basic features of Grassmann's Ausdehnungslehre
and Cauchy's "calculus of keys" - points out problems associated with simultaneous discovery in mathematics . (Belhoste 1991 236-37 and notes) .
Such simultaneous discovery is usually explained by appealing to an ide a
of the "natural" character of the object or representation in question . This
appeal to what is natural, however, not only obscures the actual process o f
discovery, it overlooks the historical fact that what is natural to one observe r
may be not at all natural to another .
In the context of quaternions, the geometrical representation is certainly key .
Indeed, Hamilton was in part motivated to develop quaternions in order
to extend the geometrical properties of complex numbers in the plane t o
3-dimensional space (See Hankins 1980, Chapter 21) . Hankel's account of
the theory of quaternions is based on the usual algebraic basis but give s
a geometric interpretation which is quite his own . He is very proud of hi s
economical account of the theory, and his remarks comparing his own wor k
to Hamilton's are informative :
The theory itself he [Hamilton] then presented with some application s
in the very comprehesive "Lectures on Quaternions" [ . . . ] [of 1853],
MfM 46:2
25 9
in a way which is extremely uncongenial to continental mathemati cians but which for the English is, so it seems, completely natural .
The theory is broken into scattered pieces, problems are not treated i n
their generality, but rather in special cases, then interrupted by appli cations and other investigations ; so that only later, and at times onl y
incidentally, are they solved fully.' (Hankel 1867, 194-196 )
He continues :
As I wished to transplant this theory onto German soil, it was neces sary to alter the presentation totally .2 (Hankel 1867, 196 )
Hankel's change in the presentation of the work consists in the use of a muc h
stricter axiomatic style . There is no discursive content about the comparative merits of the quaternion approach, and applications are kept to a
minimum. Thus Hankel claims that he has cut down Hamilton's original
545 pages to his account of about 50 pages with the sacrifice of only a
few applications . We note, however, that his geometrical interpretation o f
quaternions is markedly different from that of Hamilton . In particular it i s
much more restricted, treating them as motions which bring into coincidenc e
arcs on the surface of a sphere .
Hankel noted specific differences between the German approach and that o f
the British, insisting that Hamilton's work unaltered could not take root i n
German soil, indeed on the continent at all . This view is borne out by the
French reception of quaternions, which received their first serious discussio n
in France via the thesis of Charles-Ange Laisant in the late 1870s and hi s
subsequent books .
3 .2 Franc e
Despite the limited interest in quaternions as such, the French did take u p
the question of the priority claim to geometric representation of comple x
" Die Theorie selbst hat er dann mit einigen Anwendungen derselben in den sehr um fangreichen "Lectures an Quaternions [ . . . ]" [of 1853], in einer den continentalen Mathematikern sehr unbequemen, den Engländern aber, wie es scheint, durchaus natürliche n
Weise dargestellt : Die Theorie ist aufgelöst in zerstreute Strecke, die Probleme werde n
nicht in ihre Allgemeinheit, sondern zunächst in speciellen Fälle behandelt, dann unter brochen durch Anwendungen und andere Untersuchungen, um erst später, zuweilen nu r
gelegentlich in ihren ganzen Umfang erledigt zu werden [ . . . ]
2 Wollte ich diese Theorie auf deutschen Boden verpflanzen, so war es notwendig, di e
Darstellung total zu verändern .
260
T. Archibald
MfM 46 :2
numbers . A few years after the appearance of Hankel's monograph, in 1874 ,
Argand's 1806 treatise was republished in France under the editorship of J .
Hoüel . The book also contained the texts of the papers in the Annales d e
Gergonne by Servois, Argand, and Français which had served to make publi c
the result of Argand and establish his priority. The appearance of the book
attests to the perceived national interest in Argand's work GauthierVillars was notorious for business acumen, so the market must have at least
seemed to be solid . In a preface, Hofiel outlined the story of discoveries and
rediscoveries, citing Hankel to indicate that his (and Cauchy's) claim fo r
Argand 's priority was not merely a nationalistic one, but was accepted i n
Germany. At the time (1874) this was necessary as the aftermath of th e
Franco-Prussian War had seen a number of unfortunate incidents in whic h
allegations of plagiarism were traded back and forth between agitated Frenc h
and German nationals . (See Dugac, 1984, passim) .
4 Gauss, Klein and Sommerfeld
The text that roused Tait's ire was a report by Klein on the progress of th e
collected works of Gauss, which at first was available only in the Göttinger
Nachrichten but which was republished in the Mathematische Annalen in
1899 (Klein, 1899) . Specifically, Klein asserted that the foundations of th e
theory of quaternions appeared in an unpublished note of Gauss which ha d
been dated by the editor (Paul Stäckel) to 1819 . The key point, and the point
of subsequent contention, concerned what Klein meant by the foundation s
of the quaternion theory. His description makes this reasonably clear :
[ . . . ] already in 1819 he represented the mutations of space (as h e
calls them) by the same four parameters as used in the quaternion theory [ . . . ] [where by mutations of space he means] rotations of spac e
about the origin of coordinates, combined with an arbitrary similarit y
transformation likewise based at the origin . He designated the concept
of these four parameters as a "mutation scale", and gives explicit for mulas for the composition of two scales (thus the multiplication of tw o
quaternions), in which context he used the symbolic notatio n
(abed)(crßryb) = (ABCD )
and expressly remarked that one is here dealing with a non-commutative process! 3 (Klein 1899,130-131)
3 [ . . . ] er hat die Mutationen des Raumes (wie er es sagt), d .h . die Drehungen des
MfM 46 :2
26 1
To assess Klein's remarks, we first examine the Gauss fragment in question . It should be borne in mind that Tait's later remarks rest on Klein' s
description, rather than on the fragments themselves .
Gauss's few notes on the question appear in two parts, the first of whic h
Stäckel, who was the editor of this portion of the Werke, dated to 1819, wit h
the exception of one brief note apparently of later date which he ascribed
to 1822 or 1823 . The second part is on a separate slip which he describe s
merely as somewhat later . In the first of these rather sketchy notes, Gaus s
appears to be thinking about generalizing the mathematical description o f
rigid body motions to include stretches . He expresses such transformation s
of space coordinates (as he puts it) in terms of parameters a,b, c, and d,
the Transformationsscale in his terms, which specify such a transformation .
The second fragment specifies that k = \/a2 + b2 + c2 + d2 is the stretching factor and that if r = \/b 2 + c2 + d2 then the direction cosines of th e
rotation are b/r, e/r, and d/r. This is implicit in the first fragment . He further notes that if we consider the composition of two transformations wit h
parameters a, b, c, d and a, 0, y, 6 respectively, then this composition i s
exactly what Hamilton would later call the quaternion product of the two.
He specifically notes that the order of the composition is not indifferent, i n
fact using the notation that (a, b, c, d) (a, /3, y, 6) is not to be interchange d
with (a, ,@,-y, b) (a, b, c, d) . (Gauss, 1819, 360) . Missing in Gauss's notes i s
any account of the additive quaternion group, as we would call it today.
Tait's counterargument in support of Hamilton rests on the view that thi s
conception of quaternions does not capture the richness of Hamilton's idea s
about these objects . Nevertheless, the Gaussian fragment does very nicel y
prefigure Klein's own account of quaternions, which appeared in the first
part of his joint work with Sommerfeld on the mechanics of the top . This
Raumes um den Coordinaten-Anfangspunkt, verbunden mit einer beliebigen von letztere n
auslaufenden Aehnlichkeitstransformation bereits 1819 durch dieselben vier Paramete r
dargestellt, welche die spätere Quaternionentheorie benutzt ; er erzeichnet den Inbegriff
dieser 4 Parameter als Mutationsscala und giebt die expliciten Formeln für die Zusammensetzung zweier Scalen (also die Multiplication 2 Quaternionen) wobei er die symbolische Schreibweise
(abcd) (nßyb) = (ABCD)
benutzt und ausdrücklich bemerkt, dass es sich dabei um einen nicht commutativen Proze ß
handelt!
262
T. Archibald
MfM 46: 2
long book contained an "Exkurs über die Quaternionentheorie" which appeared as a short section at the end of the first chapter . This book had grow n
out of lectures given by Klein in 1895/96, and was originally intended t o
foster aims which Klein, writing in the third person in his preface, gave a
specific national colour: However, in contrast to Hankel's rather harsh rejection of the expository and theoretical chaos of Hamilton's treatise, Klein
diplomatically emphasized the richness of the subject for mechanics :
[In the lectures] first of all he [Klein] undertook to mention the direc t
expression of mechanical problems widespread in England, in contrast
to the more abstract colour of the German schools . On the other han d
he set out to render fruitful the methods of Riemann's function theor y
which were developed in Germany .4 (Klein and Sommerfeld 1897-1910,
III)
It was only subsequently that Klein integrated applications into the published work, in part due to the influence of Sommerfeld, who had by that
time taken a position involving teaching of mechanics . The book itself appeared in four parts, in 1897, 1898, 1903 and 1910 repectively, and thes e
exhibit a change in authorial standpoint corresponding to a shift towards
interest in applications and to the appeal to a wider readership . Thus some
of the theory set up at the beginning of the book - for example the introduction of quaternions - is not really used in what follows . Nevertheles s
we'll examine the view of quaternions given by Klein and Sommerfeld with
the end in mind of understanding Tait's objection to their approach, an d
Klein's interest in the Gauss fragment . Though the work is jointly authored
I will perhaps unfairly refer to the synthesis presented as Klein's, mostly for
abbreviation .
Klein begins by noting that despite the age of quaternions - he dates them
to Hamilton's book Lectures on Quaternions of 1853 - they remained controversial, in his view due to the metaphysical language in which Hamilto n
couched his presentation . This, says Klein, obscured the "simple geometrical
meaning" which can be given to the operations of this calculus . He hopes to
clarify this, the more so as he will show that quaternions are, as he puts i t
4 Unternahm er es in erster Linie, die namentlich in England verbreitete unmittelbarere Auffassung der mechanischen Probleme gegenüber der abstrakterer Färbung de r
deutschen Schule zu betonen, andererseits die namentlich in Deutschland ausgebildete n
Niethoden der Riemannschen Funktionentheorie für die Mechanik fruchtbar zu machen .
MfM 46:2
26 3
"a special case" of the vector calculus . [Klein and Sommerfeld 1897-1910 ,
55]
In his initial efforts to describe the top and its motions, Klein introduces
parameters which are his first example of a quaternion quantity. He considers a rectangular coordinate system xyz, thought of as fixed, and another ,
XYZ, which moves with the rotating top but which is fixed with respect t o
it . The orthogonal transformation from one coordinate system to anothe r
is then specified in the usual way by the 3 x 3 matrix of direction cosines .
These nine parameters are redundant, and may be reduced to three (sinc e
the top is rigid) . The classical way of doing this involves the use of the
Euler angles Ø, 'IP, and 0, defined as follows . The nodal line (Knotenlinie) is
defined as the line through the common origin of both coordinate system s
which is perpendicular to the plane of the vertical (the z-axis) and the axi s
of rotation (the Z-axis) . The angle Ø is then the angle between the noda l
line and the positive X-axis, -0 is the angle between the nodal line and th e
positive x-axis, and finally Ø is the angle between the z-axis and the Z-axis .
A routine calculation thus gives the matrix of the transformation from th e
XYZ to the xyz-system in terms of these Euler angles .
This familiar description is not the most useful for the top, a result of th e
fact that elliptic functions are involved in Klein's eventual description . To
obtain better parameters for his purposes, Klein then introduces comple x
variables in each of the horizontal planes, so that takin g
= (x +iy, - x-I-iy, -z )
and
(B; H, Z) = (X +iY, -x +iY, -Z)
he gets the transformation matrix in the for m
2a/ 2,y8
a-y /b ary + /S
a2 /2
ry e S2
where
a = cos (0/2) e
zcØ+~v ~
2
,
264
MfM 46 :2
T. Archibald
with similar expressions for the other parameters. These parameters have
various computational advantages arising from the fact that ab - ßy = 1.
The complex parameters may in turn be expressed in terms of real ones :
a =D+i C
,C3=-B +iA
-y =B +i A
6=D-iC .
In this notation the matrix of the change of coordinates become s
D2 + A2 - B2 - C2
2(AB+CD)
2 (AC - BD)
2 (AB - CA)
2 (AC + BD )
D2 -A2 +B2 -C2
2(BC-AD)
2 (BC + AD)
D2 + A2 + B2 + C2_
The quantities A, B, C and D satisfy the relation A2 +B2 +C 2 +D 2 = 1, and
are termed quaternion quantities by Klein . (Klein and Sommerfeld 18971910, 19-22) .
We see that these do not appear to be quaternions in the ordinary sense o f
today. They are however in the sense of Klein, who regards a quaternio n
as a geometric transformation called a Drehstreckung : the product of a rotation and a similarity. In the transformation just described, if the angle s
between the axes of the XYZ and those of the xyz systems are a, b, and c
respectively, and if w is the amount of the rotation, then we fin d
A = sin
2
cos a,
B = sin
2
cos b, C = sin
2
cos c and
D = cos
2.
Generalizing from this, Klein notes that if we choose any four parameters
and insert them in a matrix like the matrix of transformation just given ,
the result is a Drehstreckung with stretch factor A2 + B2 + C2 + D2 = T.
Thus from any system of four values we can get a transformation, and any
transformation of this kind can be described by four such parameters . [Klei n
and Sommerfeld 1897-1910, 55-56] .
Klein then introduces Hamilton's usual notation and the terminology tha t
T is the tensor of the quaternion, D its scalar part, and Ai + B j + Ck
its vector part . Klein notes in particular that it has the character of a
vector since it has a specified "axis" given by the angles a, b, and c and a
MflvI 46 :2
26 5
specified length -VT sin In this sense, Klein notes the "quaternionic" (i n
his meaning) interpretation of a pure vector : it has scalar component D = 0 ,
so the angle of rotation must be 7r, accompanied by an arbitrary stretch ( a
Wendestreckung in his terminology) .
Passing to the algebra of quaternions, Klein derives the usual multiplication from the identification of a quaternion with a Drehstreckung. This i n
turn implies the identities among the quaternion units (thus far treate d
as irreducible symbols), as well as non-commutativity . He then continues
with the vectorial interpretation, noting the position of the scalar produc t
(Grassmann's inner product) and the vector product (exterior product) i n
the quaternion product and giving the standard geometric interpretation s
of both .
We note that Klein's quaternions are very much the Gaussian objects, a s
even a casual inspection of Gauss's text shows . The Drehstreckung is exactly what Gauss sought to describe, and modulo some small changes i n
notation his results agree completely with those of Klein . It is hardly necessary to observe that in the Germany of the late nineteenth century on e
could scarcely hope for a better pedigree for one's work . It is not clear to
me at this time whether the two are independent . Klein's announcemen t
of the Gauss discovery is certainly subsequent to the lectures, as is Klein' s
involvement with the Gauss edition which occurred only after the death o f
Schering .
5 Tait's Criticism of Klei n
Tait's complaint was to rest on the fact that the view of quaternions presented by Klein misses their essence . He emphasized various fundamental
differences between the true quaternions of Hamilton and the Drehstreckungen of Klein . He underlines two aspects of quaternion theory : the invention
(as he terms it) of quaternion algebra and the discovery of the rich quaternion calculus spanning the last half of the nineteenth century . Both of thes e
are, in the view of Tait, entirely due to Hamilton .
Arguing in his usual polemic tone, Tait firmly rejected Klein's conclusio n
that a Drehstreckung is exactly the same as a quaternion :
Unfortunately [ . . . ] a Drehstreckung is [ . . . ] a totally different kind
of concept . It is obviously only a very limited form of linear and vector
266
MfM 46 : 2
T. Archibald
operator (kinematically a strain) depending on four quantities instea d
of the usual nine ; and might [ . . . ] have been designated by the name
quaternion had the name not been already more worthily bestowed .
(Tait 1899-1900, 19) .
A further problem with the Drehstreckung is the fact that the set of thes e
motions is not closed under addition .
What then is a quaternion in Tait's view? As he puts it, i t
expresses the relation of one vector to another, or supplies the facto r
required to turn one into another . (Tait 1899-1900, 19) .
For Tait, then, a quaternion is formally a vector quotient, where by a vector
is meant a free directed line segment . Completely determined by a pair o f
vectors, the quaternion is "as real as either" (ibid, 19) . This is the essenc e
of the great discovery to which Tait referred, which is accompanied by the
following representation . The pair of vectors (if distinct) determines a plane ,
an angle and has a certain ratio of lengths . This leads to the expression
q = ß= b (cos A
a
a
+ E
sin A)
where a, are the vectors, a and b their lengths, A the angle they determine ,
and e the unit normal to the plane .
The difference between this and a Drehstreckung may then be summed u p
by noting that when a quaternion is applied to a vector in or parallel to
its plane, the result is indeed a rotation combined with a change of length .
However, when the vector is not perpendicular to e, the operation still works ,
producing another quaternion, not merely a vector . Hence in Tait's analysis ,
a quaternion has no direct connection with rotation, though "as an orga n
of expression capable of dealing with all space-problems it may be used t o
describe a rotation" . Klein's Drehstreckung, "like everything else in space ,
can be represented by means of quaternions, but as a quaternion operator,
not as a quaternion ." (ibid . 20-21 . Emphasis in original) . This is becaus e
a Drehstreckung is not what Tait terms a "space-reality" . It rather needs
something to operate on, or as Tait puts, it, it "requires a subject before it
can attain embodiment ." (ibid. p. 20) .
Tait elaborates on this for some pages, displaying in various calculation s
the greater simplicity (and in his view "reality") of the quaternion . The
MfM 46:2
267
modern reader may feel that at times rather subtle distinctions borderin g
on the theological are being insisted upon . For example, emphasizing his
point that Gauss, had he possessed the quaternion concept fully, woul d
surely have discussed addition, Tait elaborates on the impossibility of addin g
Drehstreckungen as one adds quaternions :
To add two Drehstreckungen they first must be embodied, separately,
in any common vector, and the resulting vectors geometrically com pounded . Then the Drehstreckrung which produces the resultant must
be found . (ibid . 23, emphasis in original) .
6 Conclusio n
What was the effect of Tait's response? There is more than one answer . A s
far as quaternions proper were concerned, we see some acknowledgement o f
Tait's remarks in the relevant articles of the Encyklopddie der mathematischen Wissenschaften . Writing in 1898, E . Study repeats the claim in Klein' s
paper in the first volume, before Tait's claims were made (Study 1898 p .
183) . H . Rothe's article in the third volume accepts Valentiner's claim of th e
priority of Wessel over Argand, but further states of Gauss that he had th e
multiplication theorem for quaternions in 1819 or 1820 - a more restricte d
claim than that of Klein ; he furthermore notes Tait's article, though he doe s
not comment on its content .
By 1905, however, Study was more measured in his comments . In (Study,
1905, 423) he claims that Gauss had the multiplication theorem, but furthe r
remarks :
With this, however, he was far from possessing the entire theory o f
Quaternions, for which among other things the association of vector s
with simple figures in space - pairs of vectors is essential . We
must concur with English authors when they seek to treat Hamilto n
as the true discoverer of the quaternion theory after [the discovery
of Gauss' work] as well as before . The further assertion that ther e
is a difference of principle between Gauss' four-tuple and Hamilton' s
quaternions seems to us unconvincing .5 (Study, 1905, 423 )
5 Damit besaß er aber bei weitem noch nicht die ganze Quaternionentheorie, für di e
unter anderen die Zuordnung der Quaternionen zu einfachen Figuren im Raume paare n
von Vektoren - wesentlich ist . Wir werden englischen Autoren recht geben müssen,
268
T . Archibald
MfM 46 : 2
Study thus yielded to the view of Tait that Gauss lacked the full power o f
the quaternion theory, while nevertheless holding out in support of Klein' s
view, and against Tait's, that Gauss and Hamilton were not so very differen t
in their ways of thinking .
There is another aftermath, however, one which would take us too far afiel d
to explore in detail, but which should be mentioned . This is in the context
not of quaternions but of vector calculus, where many of the most important
quaternion notions eventually came to roost . Recasting quaternion notions
gave further opportunities to examine their roots, For example, we find
Emil Jahnke in 1905 tracing vectorial analysis to the barycentric calculu s
of Möbius - again a German priority over Hamilton, this time a publishe d
one . Quaternions were eventually marginalized completely as vector calculu s
became the norm, pushing debates concerning their priority well into th e
background .
References
Argand, Jean Robert (1806/1874) . Essai sur une manière de représenter les quantités
imaginaires dans les constructions géométriques . Ed . J . Hoüel . Paris : Gauthier Villars .
Belhoste, Bruno (1991) . Augustin-Louis Cauchy : a Biography . New York : Springer .
Crowe, Michael (1967/1985) . A History of Vector Analysis. New York : Dover, 1985 .
Dugac, Pierre (1984) . "Lettres de Charles Hermite à Gösta Mittag-Leffler" . Cahiers d u
séminaire d'histoire des mathématiques 5, 1984, 49-285 .
Gauss, C . F . (1819) . [Mutationen des Raumes] . Werke 8, 357-362 .
Gauss, C . F . (1831) . Anzeige to his paper "Theoria residuorum biquadraticorum, cornmentatio secunda" . Werke 2, 169-178 .
Hankel, Hermann (1867) . Theorie der Complexen Zahlensysteme insbesondere der gemei nen imaginären Zahlen und der Hamilton'schen Quaternionen . Leipzig: Voss .
Hankins, Thomas (1980) . Sir William Rowan Hamilton . Baltimore : Johns Hopkins University Press .
Jahnke, Emil (1905) Vorlesungen über die Vektorenrechnung, Leibzig, Teubner .
Klein, Felix (1899) . "Ueber den Stand der Herausgabe von Gauss ' Werke" . Math . Ann .
51, 128-133 .
Klein, Felix and Arnold Sommerfeld (1897-1910) . Ober die Theorie des Kreisels . Teubner :
Leipzig .
wenn sie nach wie vor Hamilton als den eigentlichen Erfinder der Quaternionentheori e
betrachtet wissen wollen . Unzutreffend scheint uns indessen die weitergehende Behauptung, dass zwischen den Gauss'schen Grössenquadrupel und den Quaternionen Hamilton s
ein prinzipieller Unterschied vorhanden sei .
MfM 46 :2
26 9
Knott, C . G . (1899-1900) . " Professor Klein's View of Quaternions ; a Criticism" . Proc .
Royal Soc . Edinburgh, 1899-90, 24-34 .
Kuhn, Thomas (1959) . "Energy Conservation as an Example of Simultaneous Discovery" .
Reprinted in The Essential Tension . Chicago : University of Chicago Press, 1977 ,
66-104 .
Rodrigues, Olinde (1840) . "Des lois géométriques que régissent les déplacements d'u n
système solide dans l'espace [ . . . ]" . J. math . pures appl . 5, 380-440 .
Study, E . (1898) "Höhere complexe Grössen" . Encyklopädie der mathematischen Wis senschaften. Leipzig : Teubner . I A4, 147-183 .
Study, E . (1905) . "Sir W . R . Hamilton" . Jour. D .M.V. 14, 421-424 .
Tait, P . G . (1868) Sketch of thermodynamics . Edinburgh : Edmonston and Douglas .
Tait, P . G . (1899) "On the Claim recently made for Gauss to the Invention (not th e
Discovery) of Quaternions" . Proc. Royal Soc . Edinburgh, 1899-90, 17-23 .
MfM 46:2
27 1
On the Role of Imaginary Elements i n
19th-Century Geometr y
David E . Rowe *
Our present-day knowledge of the principal research trends in geometry dur ing the nineteenth century may aptly be characterized as fragmentary and ,
in many resepects, thin . Reasons for this are not far to seek . Some mathematicians might be inclined to the ungenerous view that those few classi c
results which survived the conceptual upheaval that gave birth to moder n
mathematics during the early decades of this century constitute the vita l
core of the previous century's accomplishments . Historians of mathematic s
would presumably take a more liberal stance, but they, too, have done rathe r
little to illuminate the vast body of geometrical knowledge that accrued dur ing the period spanned by two nearly forgotten giants, Gaspard Monge an d
Sophus Lie . If modern mathematicians often take comfort in knowing tha t
today's standards in geometry are infinitely more rigorous than the one s
that guided the best work of the mid-nineteenth century, historians shoul d
be especially sensitive to the kinds of conceptual and technical problem s
that preoccupied the geometers of the distant period .
Given the sparsity of historical studies on classical nineteenth-century geom etry, we might well wonder how one can hope to achieve an overview of thi s
vast terrain . One very useful source for scanning the literature on geometri cal research during the nineteenth century was published by Duncan Sommerville in 1911 under the title Bibliography of Non-Euclidean Geometry,
including the Theory of Parallels, the Foundations of Geometry, Space of n
Dimensions . [34] This meant non-Euclidean geometry in the broad sense . In
fact, the territory Sommerville covered might well be regarded as an indicator of the breadth of activity that served to undermine the once-dominan t
* Fachbereich 17, Mathematik, Johannes Gutenberg-Universität Mainz, Postfach 3980 ,
Staudingerweg 9 (Bau 2/413), D-55099 Mainz, Germany .
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paradigm of Euclidean geometry . The so-called non-Euclidean revolution,
however, has normally been understood, first and foremost, as a supplant ing of the Kantian view that physical space was a priori Euclidean by a
doctrine that at least recognized the remote possibility that space could be
curved . i Yet beyond the immediate space problem, an issue that first really
reared its head with the work of Beltrami and Helmholtz in the late 1860s ,
we encounter in Sommerville's bibliography a host of geometrical investigations written decades earlier that no longer fit into the traditional Euclidea n
mold . Based on this information alone, it would seem quite plausible to argue, as Erhard Scholz has done, that a growing familiarity with projective
properties, higher-dimensional spaces, and the geometries associated wit h
higher analysis played a major part in weakening the privileged position o f
Euclidean geometry. 2
Running alongside these developments, geometrical practitioners took various sides in a complex, but long-standing methodological debate . This takes
us to the heart of the topic I will be concerned with here, namely the role o f
imaginaries in geometry. From nearly the beginning of the nineteenth century, imaginary points and other constructs played a central, yet problemati c
role in numerous geometrical investigations . This issue carried profound implications for the conduct of research, and not only because it reinforced th e
inner tension between those who advocated an analytic as opposed to a synthetic style . As we shall soon see, even the leading lights for the promotio n
of synthetic geometry, figures like Steiner and Staudt, knew very well that
they could not dispense with imaginaries altogether . Their strategy, rather ,
was to make these "ghostlike" entities accessible to "Anschauung", which
for Staudt meant interpreting them by means of real constructions carrie d
out in Euclidean (or affine) space . Thus, while Sommerville's bibliograph y
sheds considerable light on work that undermined Euclidean geometry b y
exploring new theories of parallelism or the geometries of higher-dimensiona l
spaces, etc ., his list ought to be supplemented by a number of other publications that document the intense efforts of nineteenth-century mathematicians to grapple with complexified geometries, or at least ones whic h
gradually made room for imaginary elements . Naturally, in this essay I can
only touch upon the relevant literature, but I do wish to leave you wit h
-For an account of how this space problem played out in Britain, see [28] .
2 For an illuminating discussion of the currents that gradually undermined the Euclidean paradigm, see [33], 342-345 .
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Imaginary Elements in 19th-Century Geometry
27 3
an impression of its larger importance . This aspect of classical geometr y
constitutes, in my opinion, one of the central themes that accompanied th e
non-Euclidean revolution when understood in its broadest terms . Indeed ,
one can follow the role of the same theme of complexification and realit y
relations in the context of the geometrization of relativity theory beginnin g
with the work of Minkowski . 3
1 Imaginaries as Ghosts : Poncelet and Steiner
Standard histories of mathematics have often overlooked the role of imaginaries in nineteenth-century geometry, in part because these accounts ten d
to portray the work of Victor Poncelet (1788-1867), Joseph Diaz Gergonn e
(1771-1859), and Michel Chasles (1793-1880) largely as a revival of syntheti c
geometry, a view promoted by both Poncelet and Chasles themselves . 4 This
interpretation can be quite misleading, however, for it tends to overloo k
how much of this French work was directly inspired by methods taken fro m
algebraic analysis . Indeed, contemporaries often referred to the approach o f
Chasles and others as a méthode mixte, a style characterized by flip-floppin g
between algebraic and synthetic techniques .
Euler's work already contributed to a widespread recognition of the utilit y
of imaginaries in algebra and algebraic analysis during the latter half of th e
eighteenth century . A cornerstone property of the complex numbers, namely
the fundamental theorem of algebra, was given by Euler but without adequate proof, so that both D'Alembert and Gauss sought to remedy this
flaw afterward . A still more general algebraic result was Bézout's Theorem ,
which originally arose in the context of algebraic elimination theory . Fro m
a geometrical standpoint, this theorem simply asserts that two (irreducible )
algebraic curves Cm and Cn given by polynomial equations P(x, y) = 0
and Q(x, y) = 0 of degree m and n, respectively, will, in general, intersect in exactly mn points when due account is taken of their multiplicitie s
(Cm, fl C, = mn points) . Some of these points of intersection may, of course ,
be imaginary . Maclaurin had already conjectured this result in 1720, an d
Euler handled various cases where imaginary roots arise . Bézout was able
3 See the discussion in [42] .
4 Both, however, stressed the decided superiority of modern methods over those o f
classical Greek geometry . See, for example [3], 268-269 .
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D.E. Rowe
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to prove that if Cm and Cn have no common asymptotic directions, then
the polynomial equation in x that one obtains by elimination methods wil l
always be of degree mn .
Figure 1 : Conics illustrating Bezout's Theorem for
rra
= n - 2.
The significance of this result for geometry was undoubtedly very clear t o
Victor Poncelet, who used imaginary points freely, introducing I, J, the so called circular points at infinity . As the simplest example of Poncelet's bol d
treatment of imaginaries, consider a pair of intersecting circles given by th e
equations x2 + y2 + ax + by + c = 0 and x2 + y2 + a'x + b'y + c' = O . Clearly,
these meet on the line Q, known as the radical axis, given b y
(a
-a')x +(b -b')y+(c -c')=0.
Noticing that the same algebra leads to a real radical axis even when th e
circles fail to intersect, Poncelet merely indicated that the vanishing points
of intersection were imaginary points that must lie somewhere on the radical axis of the two circles . His argument was based on continuity : if two
intersecting circles are gradually pulled apart, their points of intersectio n
slide along the radical axis until they eventually coincide at which point
the circles touch one another and thereafter they become imaginary. Th e
crucial issue for him was to uphold the legitimacy of his principle of continuity as a heuristic device in geometry. In the face of harsh criticism b y
France's leading analyst, A. L . Cauchy, Poncelet argued that his syntheti c
methods could be employed with the same freedom that the algebraist en joys in manipulating equations . In his Traité des propriétes projectives des
figures of 1822 Poncelet wrote :
. . . would it not be legal to accept the continuity principle in its tota l
generality in theoretical geometry, just as one has accepted it firs t
in algebra and then in the applications of this calculus to geometry,
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27 5
Figure 2 : The radical axis contains two of the four points of intersection of tw o
circles .
as a means of discovery and invention if not of proof? ([27], xiv, m y
translation . )
Just how problematic Poncelet's techniques were can be seen by returning t o
the case of circles considered above . The algebra of Bézout's Theorem lead s
us to conclude that two circles, just like any two conic sections, being curve s
of the second degree must have four points of intersection and not just two .
This led Poncelet to contemplate systems of concentric circles, which can b e
viewed as analogous to systems of rectangular hyperbolae . Since his mod e
of argumentation was synthetic, Poncelet was necessarily obscure about the
nature and properties of imaginary points, but by the mid 1800s Frenc h
geometers had grown accustomed to such arguments, enough so that thes e
"ghosts" seem not to have haunted them terribly much . Indeed, imaginarie s
became part of the "standard equipment" of algebraic geometers, and littl e
concern was shown as to their precise ontological status . For the French
school it apparently sufficed that one could visualize the most importan t
imaginary constructs by employing an appropriate imaginary transformation .
Thus, for example, the circular points at infinity could be "seen" by employ ing an appropriate imaginary transformation on the points in the projectiv e
plane . Poncelet argued synthetically, but for purposes of clarity we will fin d
it convenient to employ homogeneous coordinates (x, y, z), where z = 0 rep resents the line at infinity (homogeneous coordinates were routinely use d
by both Plücker and Möbius in the 1830s) . If we consider the image of th e
family of concentric circles centered at the origin, {x 2 + y 2 = rz 2 r E R+} ,
under the transformatio n
x' = x,
y~ =
z' = z,
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46:2
the resulting curves form the family of rectangular hyperbolae {x'2 - y'2 =
rz j2 r E III+ } whose asymptotes make an angle of 45 degrees with th e
coordinate axes . In particular, the circle x 2 + y 2 = 0 goes over into the
hyperbola xj2 -y' 2 = 0, which shares the same points of tangency at infinity,
namely (1, +1, 0) , with the other curves in the family. Poncelet reasoned
similarly that this behavior of the hyperbolae must also hold for the famil y
of circles since the imaginary circular points (1, ±i, 0) get mapped to the rea l
infinitely distant points (1, +1,0) . Thus, the circular points at infinity coul d
be seen as lying on all circles in the plane . Poncelet further called attention
to the analogous figure in three-space, the imaginary spherical circle which
lies on all spheres .
Figure 3 : For r = 0 the isotropic lines of the family of concentric circles go ove r
into the asymptotes of the corresponding system of rectangular hyperbolae .
In this connection, one should bear in mind that the French geometers o f
the early nineteenth century never severed projective geometry from it s
Euclidean roots . Thus, Poncelet often derived projective properties from
metrical ones by employing appropriate transformations and his infamous
principle of continuity . His leading disciple, Michel Chasles, made copious
use of isotropic (sometimes called minimal) lines and the imaginary spherica l
circle at infinity. These lines arise in pairs ; in the complex projective plan e
P2 (R) they correpond to the points of the equation (x - a) 2 + (y - b) 2 = 0 ,
which factors into [(x - a) + i(y - b)][(x - a) - i(y - b)] = 0 . We see at
once that only the point P with coordinates (a, b) is real, but for any suc h
fixed P the isotropic lines will pass through the imaginary circular point s
(1, ±i, 0) .
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27 7
Chasles also cultivated the study of confocal systems of conics . Such system s
have the property that the tangents to any conic drawn from the circula r
points at infinity I, J intersect in its foci (two of which are real and tw o
imaginary), as first noted by Julius Plücker .
Figure 4 : Plücker's Theorem .
The connection between confocal conics and projective concepts only be came somewhat clearer through Edmond Laguerre's projective formul a
for the angle e determined by two lines e 1 , Q2 that meet at a point P :
0(e1, £2) = 2ti log(CR(f 1 , £2,
~ j)) where CR, is the cross ratio and e l , f j
I
Figure 5 : Laguerre's Formula .
denote the isotropic lines determined by the point P, i .e. the lines joinin g
P with I and J, respectively. The orthogonal intersection of confocal conic s
follows directly from this formula and the fact that the tangents .6, 6 at a
point P common to two confocal conics are harmonically separated by L, £j
D .E . Rowe
278
(since in this case
CR( .e i ,,e 2 ,
l ,e ,e J )
MfM 46:2
= -1) .
Figure 6 : Confocal conics intersect orthogonally .
The possibilities for developing a more rigorous, but purely synthetic foundation for projective geometry were first explored by Jacob Steiner (17961863) . During his thirty-year tenure on the faculty at the University o f
Berlin, Steiner emerged as the most influential geometer in Germany (se e
[1], 55-60) . It was no accident that he evinced skepticism when it came t o
the status of imaginary entities in geometry, calling them "ghosts" (Gespenster) .5 On the other hand, he seems to have found it impossible to dispens e
with these mysterious creatures in his own work on projective geometry.
The leading representative of the opposite, analytic approach, Otto Hess e
(1811-1874), remarked perceptively at the time of Steiner's death that th e
last part of his career :
. . . scheint dem Kampfe mit dem Imaginären gewidmet gewesen z u
sein. Es gewinnt diese Hypothese an Wahrscheinlichkeit, wenn man vo n
dem Gespenst-wie Steiner sich auszudrücken liebte-in der Ebene
und Raume hört, mit dessen Hilfe er die verborgenen Wahrheite n
enthüllte. ([10], 199 .)
Hesse further surmised that Steiner had been driven to publish many of hi s
findings without any kind of supportive arguments because his own syn 5 One is reminded here of Bishop Berkeley's retort to Newton that his fluxions were
nothing but the "ghosts of departed quantities" ([39], 338) .
MfM 46 :2
27 9
thetic methods, as set forth in his classic Systematische Entwickelung de r
Abhängigkeit geometrischer Gestalten von einander, were far too weak t o
support the far-reaching researches he undertook during the later part o f
his career . It should also not be forgotten that Steiner's foundational wor k
was far from complete when he turned to his more ambitious investigations of algebraic curves and surfaces . His Systematische Entwickelung of
1832 presented only part one of a study that had originally been projecte d
as consisting of seven parts . Thus, even in his systematic investigations ,
Steiner's work remained in torso .
In the meantime, Plücker, Möbius, Cayley, Salmon, Hesse, and others ha d
made impressive strides forward in algebraic geometry mainly by employ ing new analytic methods (homogeneous coordinates, determinants, an d
invariant-theoretic techniques) . Thus, the much-vaunted synthetic style promoted by Poncelet, Steiner, and their followers looked hard-pressed whe n
it came to matching the achievements set forth by analytic geometers wh o
could appeal to a fourth dimension to explain what was meant by imaginary points . If the synthetic geometers hoped to compete with them, the n
somehow they would have to create a theory of imaginary elements withi n
the context of Euclidean space . The problem of interpreting imaginarie s
within a purely synthetic framework eluded Jacob Steiner, but not his contemporary, Karl Georg Christian von Staudt (1798-1867), who succeede d
in creating a precise method of handling the mysterious imaginary point s
and lines that had become so indispensible to geometrical research . As Felix
Klein once commented :
" . . . bei [Jacob] Steiner sind die imaginären Größen in der Geometrie
noch "Gespenster", die gleichsam aus einer höheren Welt heraus sich
in ihren Wirkungen bemerkbar machen, ohne daß wir von ihrem Wesen
eine klare Vorstellung gewinnen können . Erst von Staudt hat . . . jene
Frage vollkommen gelöst . . . " 6
Since von Staudt, the central figure in our story, remains a barely know n
figure in the history of mathematics, I begin with a few remarks concernin g
his life and career .
6 [20], 129 .
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2 Staudt's Theory of Imaginarie s
K.G .C . von Staudt was born in the charming Franconian town of Rothenburg ob der Tauber . His parents were both descended from prominent patrician families whose ancestors had played major parts in Rothenburg' s
political and social life since the 16th century . The family home where h e
was born and raised dates from this period, and can still be seen in Rothenburg today . From 1817 to 1822 von Staudt studied in Göttingen, where h e
enjoyed the rare privilege of working closely with Carl Friedrich Gauss . 7 On
numerous occasions, Gauss reported on the astronomical observations car ried out by his young student, including those von Staudt made of Mars i n
1820, the asteroid Pallas in 1821, and particularly his findings on the path
of a comet that entered the solar system during that same year . After takin g
his doctorate in Erlangen in 1822 (apparently without having to submit a
dissertation), he taught at Gymnasien in Würzburg and then Nürnberg before his appointment as Professor Ordinarius for mathematics in Erlangen
in 1835.
Von Staudt retained this position up until his death in 1867 . By all accounts ,
he was an extraordinarily dedicated teacher in an era when scholarly aspirations ran high and neohumanist ideals dominated the German universities .
Still, Erlangen's academic environment certainly could hardly be compare d
with those in Berlin, Göttingen, or Königsberg . During the academic year
1858-59 only six students were studying for a degree in mathematics, a
rather high number in view of the fact that the entire philosophical faculty recorded only nineteen auditors . The situation during the mid-1840' s
was even worse ; then the philosophical faculty had just six students! Give n
these kinds of conditions, von Staudt clearly had little, if any, opportunity
to expose young talents to his own research efforts . Thus, like many othe r
German mathematicians of this era, he worked in quietude and virtual isolation .
After von Staudt's death his successor, the brilliant young mathematicia n
and historian of mathematics Hermann Hankel (1839-1873), remained i n
Erlangen only two years before accepting a chair in Tübingen . This enabled von Staudt's closest disciple, Hans Pfaff (1824-1872), who had bee n
außerordentlicher Professor in Erlangen since 1868, to assume his forme r
7 The following information is based largely on [25] .
MfM 46 :2
28 1
mentor's position . Hans Pfaff was the son of von Staudt's predecessor, Johann Wilhelm Pfaff, and the author of Neuere Geometrie ([26]), published
in Erlangen in 1867 . This text was the first, following von Staudt's own, t o
expound the new theory of imaginaries .
Von Staudt's two principal works were entitled simply Geometrie der Lage ,
which appeared in 1847, and Beiträge zur Geometric der Lage, the first
installment of which came out in 1856 ([35], [36]) . Both represent true mile stones in the history of geometry. In his Geometrie der Lage von Staudt
gave the first systematic synthetic presentation of the principal theorems o f
plane and solid positional geometry or, more precisely, affine geometry, sinc e
von Staudt took non-metric Euclidean space with its theory of parallels a s
his starting point . To this he appended the points at infinity where paralle l
lines intersect . These thus form a distinguished set of points, contrary to th e
purely projective standpoint which regards the so-called plane at infinity a s
indistinguishable from any other plane in projective three-space .
From the outset, von Staudt made use of topological concepts, such a s
parity or the lack thereof for curves and surfaces . Such concepts, alon g
with the projective notion of harmonic separation, enabled him to uncove r
systematically a host of general theorems without making any appeal t o
metrical properties . It would be fair to say that with Geometrie der Lage the
synthetic treatment of positional geometry emerged as a mature discipline .
The well-known textbook with the same title by Theodor Reye (1866-1868 ,
2nd ed . 1886) served to anchor Staudt's achievement . Remarkably, Staudt' s
initial conception was quite traditional and partly inspired by the Euclidea n
tradition . He began by asserting that "geometry arises from the conceptio n
of an unbounded space," and then proceeded to introduce solid figures ,
surfaces, curves and points by means of general kinematical considerations .
Thus, he observed that a moving point generates a curve, a moving curv e
a surface, and finally a body emerges from a moving surface . Von Staudt' s
style of argumentation throughout was only loosely deductive . In the main ,
he relied on what might well be called descriptive arguments that steadfastly
avoided appeals to visual evidence, i .e. diagrams or pictures . Presumably the
reader was expected to produce these himself, whether mentally or on paper ,
in the course of working through the text . This kind of minimalist approach,
aimed at forcing active engagement from the reader, can be found in man y
synthetic texts of the nineteenth century, a feature that helps account fo r
the nearly total lack of appeal this genre has today .
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Already in the 1840s Staudt had hoped to develop a synthetic treatmen t
of imaginaries based on elliptic involutions (about which more below) . But ,
as he pointed out in the preface to Beiträge zur Geometrie der Lage, he
was left stymied by the problem of how to distinguish between two comple x
conjugate points, which taken together yield the same involution . His breakthrough idea thus came sometime after 1847, and was elaborated beginnin g
with the first installment of the Beiträge in 1856 . To motivate his theory,
he alluded in the preface to a particularly nice example of the insight tha t
imaginary elements can bring to the theory of conic sections . Although he
did not mention Newton in this connection, he was undoubtedly well awar e
that Sir Isaac had given a systematic treatment of the elements needed t o
construct a conic section in Book I of his Principia . There Newton carried
out the constructions for all six cases in which any combination of five elements, whether points on the curve or tangents to it, are given . But Newton
actually began with the easier case when two foci and a third point (or tan gent line) are given, thereby implicitly raising the following question . Why
do only three elements suffice in this case? Whether or not Newton eve r
asked himself this, we can be quite certain that he failed to find the follow ing elegant answer, which depends on Plücker's insight, mentioned above ,
regarding the foci of conics . As Staudt pointed out, were we given only one
tangent, knowledge of the two foci amounts to knowing four other (imaginary) tangents, since the foci are merely the points of intersection of th e
two pairs of tangents from the circular points I, J to the curve, so that fiv e
elements are, in fact, given, thereby resolving the mystery !
The theory of involutions, which von Staudt took as the basis for his theor y
of imaginaries, already formed a standard part of projective geometry in hi s
day. Several properties of involutions play an important part, for example, i n
the work of Desargues . Simply stated, an involution of the points on a line i s
a projective mapping T for which T 2 = Id. Such a mapping can have eithe r
zero, one, or two fixed points, in which case one speaks, respectively, of a n
elliptic, parabolic, or hyperbolic involution . In the first case, an analytic
representation of the mapping T sending x -* x' is given by
(x - a)(x ' - a) + b2 = 0,
where a, b E R and x E (C . From this, one sees at once that the two (missing )
fixed points are given analytically by x = a ± bi, which accounts for th e
representation of these conjugate imaginary points by means of the above
MfM 46 :2
28 3
elliptic involution on the real line "joining " this pair .
Staudt's basic idea was thus to consider the real line £ joining two conjugat e
imaginary points P and P, and then to associate this pair of points wit h
two oriented involutions of .e having P and P as fixed points . To make his
argument as transparent as possible, we make use of the analytic presentation first given by Otto Stolz in 1871 ([38]) . Thus, given an imaginary poin t
P with homogeneous complex coordinates ((, 77, T) one writes :
= C1+ i C2,
77 =771.+
7712, 7- =Tl+2T2 .
These determine two real points Pi , P2 with homogeneous coordinates :
Pl =
(S1> 771, Ti) ;
P2
= ('2, 712, T2) •
Since Pi P2 (otherwise P would be a real point), Pi and P2 determine a
real line P1 P2 given b y
P = {Sl + A(2, 77i + A772, T1 + AT2} .
Here it should be emphasized that the parameter A E C, so the line P1 P2
passes through the conjugate imaginary point P with coordinates :
= (1 - i S2,
7 / = Th. - 277 2 , T = Tl - 2T2 .
Stolz showed that from the given point pair {P1 , P2} one can write down a
formula for the other corresponding point pairs of the involution T in term s
of a parameter A :
Pi
=
= ((1
+ A(2, 771 +
A772, T1
7127 T2) = ((1
- 1(2, 771 -
1 712, 7-1 - 7-2) •
7
((i> 7i,
Ti)
+
AT2 )
and
P2 =
The oriented involutions on P1 P2 are then produced by letting A vary. Following Staudt's convention, the point P is associated with the orientatio n
obtained by letting A increase continually ; the opposite orientation, given
by decreasing values of A, yields a representation for P .
Now von Staudt showed that the totality of point pairs {P1', PD determined
by the imaginary point P produces an involution T of the real points of P1P2 .
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The fixed points (which Staudt called fundamental points) of T are just th e
imaginaries P and P . Conversely, taking any two pairs of points {P1 , P2 }
and {Pi, P2} on a real line £ one obtains a unique involution T . The fixe d
points of T will be imaginary if and only if these two pairs overlap . Thus ,
one of the points Pi, P must lie between {Pl , P2 } and the other outside the
latter pair if the involution is to represent two conjugate imaginary points .
This novel idea would have remained nothing more than a curiosity had it
not been possible to find a suitable interpretation for imaginary lines as well .
And precisely at this juncture one can begin to appreciate the true beaut y
of von Staudt's theory : all he had to do was appeal to duality . Since, i n
the projective plane, an oriented involution T of a real line .e dualizes as a n
(oriented) involution of the pencil of real lines through a real point, it follows
that the latter yields a representation for imaginary lines by means of point s
in P 2 (R) . Analogous to the situation above, in order to correspond to an
imaginary line the pencil involution must contain no real double rays . This
results if two pairs of corresponding lines always overlap with one another .
In this new context, one can now consider the possible incidence relations
between points and lines . The following four cases arise :
1. real point and real line (the familiar situation in P 2 (R)) ;
2. real point P and imaginary line £ (since P E L r P E l it follows that
P must be identical with the vertex of the line pencil representing f) ;
3. imaginary point P and real line f (the line t must be identical wit h
the line Pp that represents P) ;
4. imaginary point P and imaginary line .e (in this case the involutio n
on .ep determined by P will be in perspective with the involution o f
the line pencil determined by t) .
Using these basic constructs, one could, in principle, not only distinguis h
between real and imaginary figurations in P 2 (cC)) but also determine which
entities within a known incidence structure were real and which imaginary .
To see how this works, we consider how von Staudt employed the pola r
relation induced by a conic C in order to distinguish the real and imaginar y
points of the conic. For this purpose, we take an arbitrary point P lyin g
on the line L . From the polar relation induced by C, one obtains the pola r
line p' corresponding to P, and with it a new point P ' given by p ' n î = P' .
Since the polar of P' passes through P, the relationship between P and P'
MfM 46:2
28 5
is reciprocal, and hence this procedure induces an involution on the point s
off .
By definition, the points of C consist of those points P that lie on their ow n
polars . This property also characterizes the points of C f. Thus, the two
points where f meets C remain fixed ; they are the fundamental points of th e
involution on f where the point pairs (P, P') satisfy P = P' . In this fashion ,
von Staudt reinterpreted the question of whether or not the point P E C i s
real or imaginary into a question about the involution induced by the pola r
system of C on an arbitrary line £ passing through P. In particular, a n
imaginary point P on the conic C was represented by an elliptic involutio n
induced on a real line f .
n
Figure 7 : The involution on f pairing P with P' is elliptic .
Analytically, Staudt's construction can be carried out as follows . For a coni c
C in P2 (C) with real coefficients given by
Ax2 + 2Bxy + Cy 2 + 2Dxz + 2Eyz + Fz 2 = 0,
A, B, C, D, E, F E III ; (x, y, z) E C3 - (0,0,0)
the polar system is:
Axx' + B(xy ' + x'y) + Cyy' + D(xz ' + x 'z) + E(yz ' + y' z) + Fzz' = O .
If P = (x' , y' , z') E L, then plugging into the polar equation one obtains a
linear equation for p'. Clearly f
= P if and only if P E C. It should b e
pointed out, again, that von Staudt employed purely synthetic methods t o
describe these constructions .
n p'
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D.E. Rowe
MfM 46 :2
Imaginaries after von Staudt : Lie and Klein
Von Staudt's achievements and their importance have been praised ove r
the years by such eminent geometers as Felix Klein, Julian Lowell Coolidg e
([6], 96-100), and Hans Freudenthal ([7]) . Still, his theory was more readily
admired from afar than it was closely emulated by leading synthetic geometers. Von Staudt's uncompromising style, while fully in keeping with th e
dictates of the trend-setting Steiner school, proved a major obstacle for th e
assimilation of his ideas . One of the first to appreciate this situation was
the Austrian mathematician Otto Stolz (1842-1905) . Thus, in 1871, Stol z
gave the first analytic presentation of Staudt's theory of the imaginary i n
[38] .8 He also introduced Felix Klein to von Staudt's ideas when they met i n
Berlin during the winter semester of 1869-70, and also in Göttingen durin g
the summer of 1871 .
By this time Klein had also become friends with the Norwegian mathematician Sophus Lie, whom he also met in Berlin . Soon afterward Klein
and Lie gained a clearer idea of the methods of French metrical geometer s
during a three-month sojourn in Paris . This French style, promulgated b y
the so-called anallagmaticians, including Darboux, Moutard, and Laguerre ,
was based on the idea of taking circles in the plane or spheres in space a s
the fundamental elements and employing appropriate coordinate systems t o
study the geometric objects that arise as envelopes of circles and spheres .
The eyelids of Dupin, for example, provided a classical model for the latter .
Lie and Klein were both deeply inspired by the possibility of finding a direc t
connection between this French tradition and the mostly German research
on line geometry that had begun with Möbius but which received its main
impetus through the work of Plücker (see [29]) . Both traditions made heavy
use of imaginary elements .
Initially, the problem of interpreting imaginary points by means of real geometric structures played a central role in Lie's mathematics . Indeed, his ver y
first work presented a new theory in which points in the complex projectiv e
plane P 2 (c) become associated with certain line congruences in P 3 (R) ([29] ,
8 Stolz wrote about the poor reception of von Staudt's work : "Die Untersuchungen
dieses scharfsinnigen Geometers, welche nach rein geometrischer Methode ausgeführt sind ,
haben, so viel mir bekannt ist, bisher in der analytischen Geometrie keine Berücksichtigun g
gefunden " [38], 417 .
Mf1VI 46:2
287
224-226 ; and [22]) . 9 This kind of "anschauliche " approach to imaginary figurations clearly had a great deal of appeal to both Klein and Lie . But wherea s
Lie soon freed himself from such concerns and came to feel entirely at hom e
with the geometry of higher-dimensional complex spaces, Klein felt drive n
to unravel the relationship between real and complex structures . In fact, he
emphasized the importance of so-called "Realitätsverhältnisse" throughout
much of his career . Moreover, Klein's pursuit of a universal framework fo r
geometry led him to recognize the significance of von Staudt's systemati c
methods, which he exploited in his well-known work on the projective basi s
for non-Euclidean geometries ([12]) .
We have noted already how Klein learned about von Staudt's researche s
through Otto Stolz .' » As a Privatdozent in Göttingen, Klein even lectured
on von Staudt's theory of the imaginary and thereby contributed to the dissemination of these ideas . One of the students who attended his lectures wa s
Ferdinand Lindemann (1852-1939) . From the latter's unpublished memoir s
we learn that this experience occasioned Lindemann to pursue von Staudt' s
theory himself and to include it in the second edition of Clebsch-Lindemann ,
Vorlesungen liber Geometrie . l l
When we hear of Klein's activities in Erlangen today, the first (and probabl y
only) notion that is likely to come to mind is the famous Erlanger Program m
that he wrote on his arrival in 1872 . Curiously enough, however, during hi s
tenure in Erlangen Klein did little to follow up on the ideas he outlined in hi s
Erlanger Programm . Instead he concentrated most of his efforts on problem s
related to the role of imaginaries in algebraic geometry . 12 As a true discipl e
of Plücker, Klein hoped to shed light on the geometrical Gestalt of algebrai c
curves and their associated Riemann surfaces . This led him to develop wha t
he called a "new type" of (projective) Riemann surface (see [14], [16], [17] ,
[18]) . The possibility of exploiting Riemann's function theory in algebrai c
geometry had been realized ten years earlier by Alfred Clebsch in [4] . Klein ,
9 Not long afterward, Gottlob Frege set forth an alternative theory for treating imaginary geometric figures in his Göttingen doctoral dissertation (see [31]) .
'
°Klein himself later acknowledged the importance of the conversations he had with
Stolz when they were together in Göttingen during the early 1870's ([19], vol. 1, 51-52) .
11
"[Klein] lud mich ein, seine Vorlesung zu besuchen, und das gab wohl die Veranlassung,
daß ich diese Imaginär-Theorie in meiner späteren Bearbeitung von Clebschs Vorlesungen mit grosser Ausführlichkeit unter Hinzufügung eigener Untersuchungen dargestell t
habe ."([23], 45 )
12 For more details, see [30]
.
288
D.E. Rowe
MfM 46 : 2
however, found Clebsch's approach unsatisfying, particularly because he ha d
not followed Riemann's lead with regard to the notion of genus, the fundamental topological invariant in Riemann's geometric approach to comple x
analysis . Yet Klein also realized that conventional Riemann surfaces were
not well suited to the purposes of algebraic curve theory, which require s
that the curves be situated in P 2 ((C) in order to exploit the advantages of
Bézout's Theorem . This circumstance motivated him to set about lookin g
for a projective analogue to conventional Riemann surfaces, which were then
constructed over C or C U {oo} . Klein used his projective Riemann surfaces
to obtain a variety of results, some of them new, regarding the number o f
real and imaginary configurations in which conics are tangent to a give n
quartic curve (see [16] and [18]) . In none of these papers, however, did h e
draw any explicit connections with von Staudt's theory of the imaginary .
Figure 8: For points P' E Int(C) one has two conjugate imaginary tangents . Klei n
viewed these, following Staudt, as representing two oriented involutions of the penci l
of real lines passing through P' . By continuity, he regarded these as correspondin g
to imaginary points on two leaves of a Riemann surface that joined together alon g
C. The result was a topological sphere, a surface of genus zero.
Nevertheless, in another paper from the mid-1870's, entitled "Uber de n
Zusammenhang der Flächen," ([15], 479-480) he made some brief remark s
that reveal how the . two theories dovetailed together very nicely. Klein' s
main concern in this paper was to give a method for determining whethe r
or not a surface was orientable and to emphasize that this was an intrin sic property independent of any embedding in space . As his principal tool ,
he introduced the notion of an indicatrix, a local pencil of oriented curve s
passing through a point P of the surface . This served to provide a local
orientation-if by moving P anywhere around the surface this orientation
remained preserved then the surface was orientable . Klein also emphasize d
the close connection between his indicatrix construction and von Staudt's
MfM 46 :2
28 9
Figure 9 : Klein's projective Riemann surface for a nonsingular cubic C with two
real components . The three real inflection points (one of which lies at infinity) ar e
transformed into cusps in passing to the dual curve C* . This dualized curve has
class 3, so that three tangents to C* pass through each point in the plane . In the
outer and inner regions of the plane three real tangents pass through an arbitrar y
point, whereas only one tangent can be found for points in the annular region . The
"missing" imaginary tangents correspond to points on two leaves of a Rieman n
surface of genus one, a topological torus .
theory of imaginary lines as represented by oriented involutions on pencils
of lines . In a projective setting, he noted that such an oriented involutio n
Ip can be localized by constructing a small conic about P whose conjugat e
diameters correspond to the reciprocal lines of Ip . This yields a directrix
that one can then move along curves of a projective surface .
Klein's key insight here lay in recognizing the compatibility of these ideas ,
borrowed from von Staudt, with his projective Riemann surfaces . The latter ,
he noted, are always orientable due to the fact that, by construction, th e
two possible orientations at a point P are realized as two distinct point s
that lie on different sheets of the surface .
Certainly much of this must be regarded as idiosyncratic mathematics, fo r
it left no discernible mark on geometrical research after the turn of the century. Nevertheless, even after the publication of Hilbert's Grundlagen der
Geometrie all]) in 1899, various theories for treating imaginary geometrical
290
D.E. Rowe
MfM 46 : 2
structures continued to fascinate a handful of geometers . 13 As the modern ,
axiomatic standpoint grew stronger, however, the ontological issues tha t
originally motivated mathematicians to investigate possible relationship s
between real and imaginary structures eventually lost their urgency . Klein's
journey may be regarded as one of the last to reach the frontiers of researc h
in algebraic geometry mounted on arguments that he could convey in a pic ture . The reliance on anschauliche approaches, which Klein often promoted ,
eventually came under sharp attack .
In 1917 Eduard Study leveled a harsh assault on the sloppy, even fallaciou s
arguments set forth by Lie and Klein in a review article published in th e
Jahresberichte der deutschen Mathematiker-Vereinigung ([41]) . Study, who
had habilitated in Leipzig during the mid 1880s under Klein's supervision ,
saw himself, and rightfully so, as the leading representative of German algebraic geometry in the tradition of Plücker and Clebsch. His was no wholesale critique of Anschauung, and to make his case for the importance o f
anschauliche arguments in geometry he even quoted Poincaré with strong
approval . Yet Study had no patience with murky thinkers like Lie and Klei n
who, to his mind, refused to recognize the importance of systematic investigations undertaken with care for the details . He considered Lie's spher e
geometry a particularly glaring case in point, but ten years earlier he als o
published a similarly devastating critique of certain results in Lie's theor y
of differential invariants ([40]) .
In the case of Klein, Study portrayed him as having betrayed the principle s
of his own Erlanger Programm. Having accompanied Klein during his tri p
to Chicago in 1893, Study had witnessed up close how his former mento r
skillfully exploited the opportunity to make propaganda for his Erlange r
Programm. He also knew full well that Klein had done very little in pursui t
of this program during the intervening twenty years . As a champion of mod ern invariant theory, Study saw his own work as fully in accord with th e
philosophy of Klein's Erlanger Programm namely that geometry shoul d
be understood as the systematic study of those invariants associated with a
given transformation group. Yet Stud y' s sweeping criticisms of Klei n's detrimental influence on geometry in Germany seem to have impressed very fe w
of his contemporaries, despite the fact that much of what he asserted woul d
have been difficult to refute . Even his leading pupils, Wilhelm Blaschke and
13 See, for example, [32] . The dissertations of two Harvard geometers, J . L . Coolidge
and William Graustein, also dealt with imaginaries in geometry .
MfM 46 :2
29 1
Julian Lowell Coolidge, were among Klein's most avid admirers, probabl y
contributing more to maintaining his reputation than they did Study's own .
At any rate, we can see from this example, as I suggested at the outset, tha t
very little is known about even the most famous protagonists of various approaches to geometry during the nineteenth century and beyond . Surely the
time is ripe for deeper and more detailed historical investigations .
Acknowledgment s
The author wishes to express his thanks to Jesper Lützen for his carefu l
reading of the manuscript . He also thanks two anonymous referees for thei r
helpful comments . He is especially grateful to Stephan Endrass for converting his crude, hand-drawn pictures into pleasing, computer-generate d
graphics.
References
[1] Kurt-R . Biermann, Die Mathematik und ihre Dozenten an der Berliner Universität,,
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en Géométrie, Paris : Gauthier-Villars, 1837 .
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Journal für die reine und angewandte Mathematik 63 (1864), 189-243 .
[5] Julian Lowell Coolidge, The Geometry of the Complex Domain, Oxford : Clarendo n
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MfM 46 : 2
[11] David Hilbert, Grundlagen der Geometrie, (Festschrift zur Einweihung des Göttinger
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(1876), 476-482, or or [19], 2, 63-67 .
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Springer, 1926 .
[21] Felix Klein, Vorlesungen über die Entwicklung der Mathematik im 19 . Jahrhundert,
2 vols ., Berlin : Springer, 1926-27 .
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[23] Ferdinand Lindemann, "Lebenserinnerungen," (unpublished memoirs from ca . 1910) .
[24] Isaac Newton, The Mathematical Papers of Isaac Newton, vol . 4 (1674-1684), ed . D .
T . Whiteside, Cambridge : Cambridge University Press, 1971 .
[25] Max Noether, "Zur Erinnerung an Karl Georg Christian von Staudt," Jahresbericht
der Deutschen Mathematiker-Vereinigung 32 (1923), 97-119 .
[26] Hans Pfaff, Neuere Geometrie, Erlangen, 1867 .
[27] Victor Poncelet, Traité des propriétes projectives des figures, Paris : 1822 .
[28] Joan L . Richards, Mathematical Visions . The Pursuit of Geometry in Victorian Eng land, Boston : Academic Press, 1988 .
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Boston : Academic Press, 1989, 1, 209-273 .
[30] David E . Rowe, "In Search of Steiner's Ghosts : Imaginary Elements in NineteenthCentury Geometry," in Le nombre une hydre à n visages. Entre nombres complexe s
et vecteurs, ed . Dominique Flament . Paris : Editions de la Maison des Sciences de
L'Homme, 1997, 193-208 .
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[32] Percey F . Smith, "On Sophus Lie's Representation of Imaginaries in Plane Geometry," American Journal of Mathematics 25 (1903), 165-179 .
MfM 46 :2
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[33] Erhard Scholz, Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré,
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York : Chelsea, 1970 .
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[37] Jacob Steiner, "Sätze über Curven zweiter und dritter Ordnung," Journal für di e
[38] Otto Stolz, "Die geometrische Bedeutung der complexen Elemente in der analytischen Geometrie," Mathematische Annalen, 4 (1871), 416-441 .
[39] Dirk J . Struik, ed ., A Source Book in Mathematics, 1200-1800, Cambridge, Mass . :
Harvard University Press, 1969 .
[40] Eduard Study, `"Kritische Betrachtungen über Lies Theorie der endlichen kontinuierlichen Gruppen," Jahresbericht der deutschen Mathematiker-Vereinigung 17 (1908) ,
125-142 .
[41] Eduard Study, "Über Lies Kugelgeometrie," Jahresbericht der deutschen Mathematiker-Vereinigung 25 (1917), 96-113 .
[42] Scott Walter, "The Non-Euclidean Style of Minkowskian Relativity," in The Symbolic
Universe, ed . Jeremy Gray, Oxford : Oxford University Press, 1999, 91-127 .

Around Caspar Wesse l and the Geometric Representation of

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